In all the preceding examples, the right ascension of the sun ought to have been calculated for the moment of the star's passing the meridian, as will be more fully explained in the precepts of Tabies XXX. XXXI. TABLE IX. Semi-diurnal and Semi-nocturnal arches.-This table exhibits half the time that a celestial object continues above the horizon when the latitude and declination are of the same name, or below when they are of a contrary name; the former time being usually called the semi-diurnal arch, the latter the semi-nocturnal arch; whence the time of rising and setting may be computed by the following rules :— To find the time of the sun's rising and setting, and the length of the day and night. RULE. Find the sun's declination at the top of the table, and the latitude in either side column; under the former, and opposite the latter, will be the time of the sun's setting if the latitude and declination are of the same name, but the time of rising if of different names. The time of rising, subtracted from 12 hours, will give the time of setting; or the time of setting, subtracted from 12 hours, will give the time of rising. The time of rising, being doubled, will give the length of the night; and the time of setting, being doubled, will give the length of The day. EXAMPLE I. Let it be required to find the time of the sun's rising and setting, with the length of the day and night, in latitude 51° north, the 19th of July, 1837. The sun's declination on the given day is 20° 52′ north, or 21° nearly, under which, and gainst the latitude 51°, stand 7h. 53m., the time of the sun's setting on the given day, in lat. 51° north, which doubled, gives 15h. 46m., the length of the day; and by subtracting 7h 53m. from 12h., the remainder, 4h. 7m., is the time of the sun's rising, which doubled gives h. 14m. the lengt of the night. But, when the sin has 21 south declination in this latitude, the time of sun-setting be Comes 4h. 7m., the time of rising 7h. 53m., the length of the day 8h. 14m., and the length of the night 15h. 46m., as was the case nearly on the 26th of November, 1837. When a great degree of accuracy is required, proportiona! parts may be taken for the minutes of latitude and declination. To find the time of rising and setting of stars whose declination does not exceed 23°28 Enter Table IX. and find the star's declination at the top, and the latitude at the side; under the former, and opposite to the latter, will be the semi-diurnal arch, when the latitude and declination are both north or both south; but if one be north and the other south, the difference between the Tabular number and 12 hours will be the semi-diurnal arch. Find the time of the star's coming to the meridian according to the precepts of Table VIII., and subtract therefrom the semi-diurnal arch; the difference will be the time of rising; or by adding together the semi-diurnal arch, and the time of passing the meridian, the time of setting will be obtained. Sum, rejecting 12 hours, is the tire of setting in the morning..... 244 17 26 12 Star sets 26 minutes after 5 in the evening.. 5 26 in like manner may the rising and setting of any planet be found when the declination does not exceed 23° 28', and the time of the passage over the meridian is known. Suppose it was required to find the time of Jupiter's rising and setting, August 7, 1836, civil account, in the latitude of 52° N. In the Nautical Almanac for 1836, I find that Jupiter passes the meridian, August 6d. 23h. 11m., or August 7d. 11h. 11m. A. M., civil account, his declination being 20° 17′ N., or nearly 20. Under the declination 20°, and opposite to the latitude 52°, stand 7h. 51m., which is half the time Jupiter is above the horizon; this subtracted from 12h. leaves half the time that he is below the horizon, 4h. 9m.; subtracting 7h. 51m. from 11h. 11m. A. M. leaves 3h. 20m. A. M., August 7, for the time of Jupiter's rising; and added to 11h. 11m. gives 7h. 2m. P. M., August 7, for the time of Jupiter's setting, nearly. Suppose it was required to find the ti.ne of the moon's setting, May 2, 1836, civil account, in the latitude of 52° N. In the Nautical Almanac, pages iv. v., we find that the moon passes the meridian May 1d. 12h. 35m., or May 2d. Oh. 35m. A. M., civil account; her declination being about 18 S. Under the declination 18°, and opposite to the latitude 52°, stand 7h. 38m., half the time the moon is below the horizon. Subtracting this from 12h. we get half the time she is above the horizon, 4h. 22m.; adding this to Oh. 35m. we obtain the time of the moon's setting May 2d. 4h. 57m., civil account. If we subtract 4h. 22m., from Oh. 35m. +24h., we get the time of rising May 1d. 20h. 13m. or May Id. 8h. 13m. P. M. If greater accuracy is required, you must find the time at Greenwich corresponding to this approximate time of her rising and setting; then find the moon's declination, and the right ascensions of the sun and moon for that moment of time. The former subtracted from the latter leaves the corrected time of the moon's passing the meridian. With these data repeat the operation. In this way we may obtain the time of rising and setting to any degree of accuracy. Instead of taking the difference of the right ascensions of the sun and moon, you may take the daily difference in the time of her coming to the meridian of Greenwich, and take a proportional part for the longitude of the place of observation (by means of Table XXVIII.) and another proportional part, for the interval between the hour of passing the meridian, and the time of rising or setting.* It may be noted, that the numbers of Table IX. were calculated for the moment the sun's centre appears in the true horizon; allowance ought to be made for the dip, parallax, and refraction, by which the sun and stars, when near the horizon, appear in general to be ele vated above half a degree above their true place, and the moon as much below her true place. TABLE X. For finding the Distance of any Terrestrial Object at Sea. The explanation and use of this table is given in Problems useful in Navigation, VIII-XII., pages 95, 96. TABLE X. A. For the planets is similar to Table XIV. for the sun. The parallax is found by entering at the top with the planet's horizontal parallax, and at the side with the altitude of the planet; the corresponding number is the parallax of the planet in altitude. TABLE XI. Table of Proportional Parts.-The method of using this table is given i the preparations necessary for working a lunar observation page 229. TABLE XII. Table of Refraction.-Explained in page 154. TABLE XIII. Dip of the Horizon.-Explained in page 154. TABLE XIV. Sun's Parallax in Altitude.-Explained in page 153. TABLE XV. Augmentation of the Moon's Semi-diameter-The moon's semi-diameter given in the Nautical Almanac is the same as would be seen by a spectator supposed to be placed at the centre of the earth, or nearly the same as would be seen by a spectator on the surface of the earth, when the moon is in the horizon. Now, when the moon is in the zenith of the spectator placed at the surface, her distance from him is less than when at the horizon by a semi-diameter of the earth; consequently her apparent semi-diameter must be aug. mented in proportion as the distance is decreased, that is, about one sixtieth part, or 16". At intermediate altitudes between the horizon and zenith, the augmentation is proportional to the sine of the altitude, and the value for every 50 or 10° of altitude is given in Table XV. The augmentation corresponding to the altitude being found in the table, must be added to the semi-diameter taken from the Nautical Almanac for the time of observation reduced to Greenwich time, as is explained in the preparations necessary for working a lunar observation. TABLE XVI. Dip of the Sea at Different Distances from the Observer.-Explained in rage 155. TABLE XVII. For finding the Difference between 60' and the Correction of the Altitude of a Star or Planet, for Parallax and Refraction; also the corresponding Logarithm.-The first page of this table is to be used for a star, or for the planets Jupiter and Saturn, whose parallax is small. In other cases, that page of the table is to be used, which contains, at the top, the horizontal parallax of the planet, or comes the nearest to it; the tables being calculated for every 5 of horizontal parallax, from 0" to 35". TABLE XVIII. For finding the Difference between the Correction of the Sun's Altitude for Parallax and Refraction and 60', also a Logarithm corresponding thereto.-The manner of taking the numbers from the two preceding tables, and the uses to which they may be applied, are explained in the preparations necessary for working a lunar observation, page 230, &c. TABLE XIX. For finding a Correction and Logarithm used in the First Method of work In strictness, this last correction, found by the table, ought to be decreased in the ratio of 24h. to 24h. iD creased by the daily difference of the time of the moon's passing the meridian. ing a Lainar Observation.-The correction found in this table, being subtracted from 59′ 42 will leave a remainder equal to the correction of the moon's altitude for parallax and refraction. It will be unnecessary here to point out the method of taking out this correction as it is fully explained in the first pages of the table. It may not however, be amiss to observe, that, after constructing the logarithms of this table, it was concluded to subtract there froin the greatest correction of the Table C corresponding, in order to render those corrections additive. Thus the logarithm corresponding to the alt. 30 and hor. par. 54' was found at first to be 2372; and for the hor. par. 54' 10 the correction was 2358; so that if these numbers had been published, the correction for seconds of parallax would have been subtractive; but as this would have been inconvenient, it was thought expedient to subtract from each of the numbers thus calculated, the greatest corresponding correction of Table C, which in the preceding example is 12; by this means the above numbers were reduced to 2360 and 2346 respectively, and the corrections of Table C were rendered additive. In a similar manner the rest of the logarithms of the table were calculated. It is owing to this circumstance that the corrections in Table C for 0" of parallax are greater than for any other number. Similar methods were used in calculating the other numbers of this table, and in arranging the Tables A and B. TABLE XX. Third Correction of the Apparent Distance.-The manner of finding the correction from this table is explained in the first method of correcting the apparent distance of the moon from the sun, page 231; and also at the bottom of the table. TABLE XXI. To reduce Longitude into Time, and the contrary.—In the first column of this table are contained degrees and minutes of longitude, in the second the corresponding hours and minutes, or minutes and seconds of time; the other columns are a continuation of the first and second respectively. The use of this table will evidently appear by a few examples. EXAMPLE II. TABLE XXII. Proportional Logarithms.-These logarithms are very useful in finding the mean time at Greenwich corresponding to the true distance of the moon from the sun or star, as 18 explained in the examples of working a lunar observation. They may be also used like common logarithms, in working any proportion where the terms are given in degrees, minutes, and seconds; or in hours, minutes and seconds, as in the example of taking a lunar observation by one observer. The table is extended only to 30 or 3h .; ano if any of the terms of a given proportion exceed 3° or 3h., you may take all the terms one grade lower; that is, reckon degrees as minutes, minutes as seconds, &c., and work the proportion as before; observing to write down the answer one grade higher; that is, you must esti mate minutes as degrees, seconds as minutes, &c. Instead of taking all the terms one grade lower, you may change two of the terms only, viz. one of the middle terms and one of the extreme terms; thus the 1st and 3d or the 1st and 2d may be taken one grade less, and the fourth term will be given correctly; but if the fourth term be taken one grade less, yon must, after working the proportion, write it one grade higher, as is evident. To illustrate this, we shall give the following examples: TABLE XXIII. For finding the Latitude by two Altitudes of the Sun.-The manner of using this table is explained in the examples of double altitudes given in pages 185-189. TABLE XXIV. Natural Sines.-This table contains the natural sine and cosine for every minute of the quadrant to the radius 100000, and is to be entered at the top or bottom will be the natural sine and cosine respectively, observing that if the degrees are found at the top, the name sine, cosine, and M., must also be found at the top, and the contrary if the degrees are found at the bottom. Thus 43366 is the natural sine of 25° 42', or the cosine of 64° 18'. We have given in nis edition of the present table, in the outer columns of the margin, tables of proportional parts, for the purpose of finding nearly, by inspection, the proportional part corresponding to any number of seconds in the proposed angie; the seconds being found in the marginal column marked M., and the correction in the adjoining column. Thus, if we suppose that it were required to find the natural sine corresponding to 25° 42′ 19′′; the difference of the sines of 25°42′ and 25° 43' is 26; being the same as at the top of the lefthand column of the table; and in this column, and opposite to 19", in the column M., is the correction 8. Adding this to the above number 43366, because the numbers are increasing, we get 43374 for the sine of 95° 42′ 19′′. In like manner, we find the cosine of the same angle to be 90108—4=90104, using the right-hand columns, and subtracting because the numbers are decreasing; observing, however, that the number 14 at the top of this column varies 1 from the difference between the cosines of 25° 42′ and 25° 43', which is only 13; so that the table may give in some cases a unit too much, between the angles 25° 42′ and 25° 43'; but this is, in general, of but little importance, and when very great accuracy is required, the usual method of proportional parts is to be resorted to, using the actual tabular difference. Similar tables of proportional parts are inserted in this edition of Tables XXVI. XXVII. for the like purpose. TABLE XXV. Logarithmic Sines, Tangents, and Secants to every Point and Quarter Point of the Compass.-This table is to be used instead of Table XXVII. when the course is given in points. The course is to be found in the side column, and opposite thereto will be the log. sine, tangent, &c.; the names being found at the top when the course is less than 4 points, otherwise at the bottom. TABLE XXVI. Logarithms of Numbers.-The explanation and uses of this table are given in the article treating on logarithms in the body of the work, pages 28-33. TABLE XXVII. Logarithmic Sines, Tangents, and Secants.-This table is explained in the corresponding article in the body of the work, pages 33 -35. TABLE XXVIII. For reducing the Time of the Moon's Passage over the Meridian of Greenwich, to the Time of her Passage over any other Meridian.-The manner of doing this is explained in the corresponding part of the body of the work, page 170. TABLE XXIX. Correction of the Moon's Altitude for Parallax and Refraction.—The mean correction of the moon's altitude is given in this table for every degree of altitude from 10 to 90o. The manner of using this table is explained in pages 172, 173. TABLES XXX. XXXI. For finding the Sun's Right Ascension and Declination, the Equation of Time, and the Moon's Right Ascension.-The uses of these tables will be seen by the following examples, the values for apparent noon being taken from the Nautical Almanac, together with the horary motions. ...... Equation of time, 1836, July 9d. 8h. 20m. + 4 52.3 EXAMPLE IV. Required the moon's right ascension in 1836, May 11d. 17h. 35m. 36s. mean time, astronomical account at Greenwich. h. m. s. By N. A. Rt. As. May 11d. 18h. is 0 55 40.89 Add to Right Asc. May 1ld. EXAMPLE VI. 1 52.17 112 17 1 06 0 53 49 nearly 0 54 55 Required the moon's declination in 1836, Sept. 10d. 8h. 20m. 30s. mean time, astronomical account, at Greenwich. Here the motion in declination for 10m. is by N. A Motion for 20m. is 2 × 140′′.07=280".14 and 30s. at side, in col. M. the correction, divided by 10, is 7 0 Motion in declina. in 20m. 30s. 287".1 = 4′ 47′′.1 EXAMPLE VII. Required the moon's declination in 1836, May 11d. 17h. 35m. 36s. mean time, astronomical count st Greenwich. Here the motion in declination for 10m. is by N. A. 143.02. Motion for 50m. is 143.02 x 3 = ......... 5m. is 143.02 x 0.5..... Tab. XXX. 143" at top, and 36s. at side in col. M. the corr. divided by 10 is 42941 Motion in declination is Add to declination May 11d. 17h. by N. A.............. D's declination May 11d. 17h. 35m. 35s.... Here the correction 8' 29".2 is added, because the declination is increasing. 509 2 8′ 29′′.2 2 27 55" 1 N. If we wish to find accurately the time that any star comes to the meridian, or the time of rising or setting, we must take the sun's right ascension for noon at Greenwich, from the Nautical Almanac; then the star's right ascension from Table VIII., and with these find the approximate time of rising, setting, or coming to the meridian, by the method already given in the precepts for using Tables VIII. and IX. Then calculate the sun's right ascension for this approximate time, and repeat the operation till the assumed and calculated times agree, and we shall have the true time required. To explain this method, we shall give the following examples: To find the time of rising or setting of a star. RULE. Enter Table IX. with the declination of the star at the top, and the latitude of the place at the side; the corresponding number will be the time of the star's continuance above the horizon, when the latitude and declination are of the same name; but if they are of dif ferent names, the tabular number subtracted from 12h., will be the time of continuance above the horizon. Add this time to the star's right ascension, if we wish to find the time of setting; but subtract the former from the latter if we wish the time of rising. From this sum or difference subtract the sun's right ascension* corrected for the longitude of the place; the remainder will be the approximate time sought. Enter Table XXXI. with the distance of this approximate time from noon, and the horary variation of the sun's right ascension: the correction corresponding is to be added to the approximate time in the forenoon, but subtracted in the afternoon, and we shall have the correc'ed time of rising and setting. Increasing the number from which the subtraction is to be made, by 24 hours, when necessary. Rejecting 24 hours when it exceeds 24 hours. If the time of rising or setting be more thau 12h., it will be after midnight; but if less than 12h., it will be before midnight. 50 |