In the Nautical Almanac for 1820, I find that Jupiter passes the meridian March 7d. 23h. 10m. or March 8d. 11h. 10m. A. M. civil account, his declination being 10° 55 S. or nearly 110. Under the declination 11°, and opposite to the latitude 52°. stand 6h. 58m. which is half the time Jupiter is below the horizon; this subtracted from 12h. leaves half the time that he is above the horizon, 5h. 2m.; this subtracted from 11h. 10m. A. M. leaves 6h. 8m. A. M. of March 8, for the time of Jupiter's rising; and added to 11h. 10m. gives 4h. 12m. P. M. March 8, for the time of Jupiter's setting. Suppose it was required to find the time of the moon's rising and setting, May 5, 1820, civil account, in the latitude of 52° N? In the Nautical Almanac, page 6, I find that the moon passes the meridian May 4d. 18h. 7m. or May 5d. 6h. 7m. A. M. civil account; her declination being about 210 S. Under the declination 21°, and opposite to the latitude 52°, stand 7h. 58', half the time the moon is below the horizon, which subtracted from 12h. leaves half the time she is above the horizon, 4h. 2m.; this subtracted from 6h. 7m. leaves 2h. 5m. A. M. the time of the moon's rising, and added to 6h. 7m. gives 10h. 9m. A. M. the time of her setting, nearly. 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 elevated 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 VII. and VIII. pages 174, 175. TABLE XI. Table of Proportional Parts.-The method of using this table is given in page 152. TABLE XII. Table of Refraction.-Explained in page 108. TABLE XIV. Sun's Parallax in altitude.-Explained in page 107. 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 augmented 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 5° 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 was explained in page 152. TABLE XVI. Dip of the sea at different distances from the observer.-Explained in page 109. TABLE XVII. For finding the difference between the refraction of a star and 60'; also a log. corresponding. 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 is explained in page 153, and the uses to which these tables may be applied are explained in pages 153 and 160. TABLE XIX. For finding a correction and logarithm used in the first method of working a lunar 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 * In strictness, this last correction, found by the table, ought to be decreased in the ratio of 24h. 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, was concluded to subtract therefrom 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 method of finding the correction from this table is explained in pages 154, 160, 162. 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 I. h. m. s. EXAMPLE II. Required the time corresponding to 50° 31'? Required the degrees and minutes corresponding to 6h. 35m. 20s.? Opposite 6h. 32m. Os. Opposite 50° in col. 1 is 31/ Sought time 24 3 22 4 1 20 6 33 20 in col. 4 is 98° 0 20 98 20 TABLE XXII. Proportional Logarithms.-These logarithms are very useful in finding the apparent time at Greenwich corresponding to the true distance of the moon from the sun or a star, as is explained in page 154. 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 examples page 163. The table is extended only to 3° or 3h. and if any of the terms of a given proportion exceed 30 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 estimate 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, you must, after working the proportion, write it one grade higher, as is evident. To illustrate this we shall give the following examples. 2' 32" 2!!! P. L. P. L. Here the 2d and 4th terms must be taken one grade less. 8.8239 As 16' 0" 1.4091 Is to 3 27 1.6185 So is S 10 1.8515 To 0' 41" P. L. 8.9488 1.7175 1.7547 2.4210 Which taken one grade higher is 2° 32′ 2′′ the Which taken one grade higher is 41', the ananswer required. TABLE XXIII. For finding the latitude by two altitudes of the sun.—' of using this table is explained in page 128, et seq. swer required. -The manner TABLE XXIV. Natural Sines.-This table contains the natural sine and co-sine for every minute of the quadrant to the radius 100000, and is to be entered at the top or bottom with the degrees and at the side marked M. with the minutes, the corresponding numbers will be the natural sine and co-sine respectively, observing that if the degrees are found at the top, the name sine, co-sine, 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 co-sine of 64° 13'. 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 page 28, et seq. TABLE XXVII. Logarithmic Sines, Tangents, and Secants.-This table is explained in page 33, et seq. 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 page 123. 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 90°. The manner of using this table is explained in page 125 TABLE XXX. For reducing the Moon's declination given in the Nautical Almanac for noon and midnight at Greenwich, to any other time under any other meridian.— The manner of using this table is explained in page 124. In addition to which it may be observed that 12h. are marked both at the bottom of the left hand column and at the top of the right hand column; but this can cause no embarrassment, because when the time at Greenwich is 12h. the declination must be taken from the Nautical Almanac for midnight, without any correction. TABLE XXXI. For reducing the Sun's right ascension in time, as given in the Nautical Almanac for noon at Greenwich, to any other time under any other meridian. This table is useful in finding the Sun's right ascension at any time, by means of the right ascension given in the second page of the Nautical Almanac for noon at Greenwich. This table must be entered at the top with the daily variation of the sun's right ascension; and in the left hand column with the given time from noon; or in the right hand column with the longitude of the place; under the former and opposite to the latter will stand a correction in minutes and seconds, to be applied to the sun's right ascension at noon at Greenwich. The correction found with the time from noon, is to be added in the afternoon, but subtracted in the forenoon; and the correction found with the longitude of the place, is to be added in west, but subtracted in east longitude. Instead of finding the correction separately for the longitude of the place and the time from noon, you may find the whole correction at one entry, in the following manner. Turn the ship's longitude into time (by Tab. XXI.) and add it to the given time when in west longitude, but subtract the longitude when east; the sum or dif ference will be the time at Greenwich; find this time in the side column" and the daily variation at the top, corresponding to which will be the sought correction; which is to be added to the sun's right ascension for the preceding noon at Greenwich. Required the sun's right ascension at noon, Required the sun's right ascension at noon, June May 24, 1820, sea account, in the longitude of 45° 24, 1820, sea account, in the longitude of 120° E. W. from Greenwich? from Greenwich? * If the time at Greenwich be more than 12h. you must first take out the correction for 12h. and The third and fourth examples may be worked by a single entry of Table XXXI. as follows. If you wish to find accurately the time that any star comes to the meridian, or the time of rising or setting, you 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 you will have the true time required. To explain this method, I shall give the following examples. To find the time when a star comes to the meridian. At what time was Aldebaran on the meridian of Jan. 2, sea account, is Jan. 1, N. A. on Aldebaran's R. A. 4h. 25m. 36s. Add 24 Difference is the approximate time h. m. s. EXAMPLE II. At what time was Poilux on the meridian of a place in the longitude of 70° 46′ W. March 31 1820, sea account? March 31, sea account, is March 30, N. 28 25 36 Approximate time of southing Now calculating the sun's 'R. A. for this time in And for the long. 70° 46′ W. of Greenw. the long of 70° 50′ W. from Greenwich, I find it was Aldebaran's R. A. + 24h. True time of coming to the merid. h. m. s. 9 39 9 b. m. s. 0 35 41 734 17 6 58 36 43 47 which subtracted from the approximate time of southing 6h. 58m. 36s. leaves the true time 6h. 56m. 49s. The method (used in the last example) of applying the corrections to the approximate time, instead of applying them to the right ascension of the sun, will be found the most expeditious; but it must be noted, that the corrections to be applied to the approximate time must have a contrary sign to what they would have when applied to the right ascension. To find the time of rising or selling 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 different 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 you wish to find the time of setting; but subtract the former from the latter if you 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.t Enter Table XXXI. with the distance of this approximate time from noon, and the daily 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 you will have the corrected time of rising or 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 than 12h will be after midnight; but if less than 12h. it will be before midnight. 息 At what time did the star Aldebaran set, May At what time did the Dog-Star Sirius rise in the 24, 1820, sea account, in the latitude of 38° 53′ N.latitude 39° 20' N. and the longitude of 76° 50 W. and the longitude of 77° W? Jan. 2, 1820, sea account? The star's declination was 16° 8' N. and the lati-The star's declination is 16° 28′ S. and the latitude tude 38° 53′ N. corresponding to which in Ta- is 39° 20′ N. corresponding to which in Tab. IX. ble IX. is 6h. 54m is nearly 6h. 56m. Which subtracted from 12 0 TABLE XXXII. Variation of the sun's altitude in one minute from noon. TABLE XXXIII. To reduce the numbers of Table XXXII. to other given intervals of time from noon. The method of using the two preceding tables is explained in pages 136 and 137. TABLE XXXIV. Errors arising from a deviation of 1' in the surfaces of the central mirror. This table shows the error arising in measuring an angle by an instrument of reflection from a deviation of 1' in the parallelism of the surfaces of the central mirror, the line of intersection of those surfaces (produced if necessary) being perpendicular to the plane of the instrument. If the line of intersection be inclined to that plane, the numbers in the table must, in general, be decreased in proportion to the sine of the angle of inclination. The second, third, and fourth columns of the table are calculated upon the supposition that the surface of the horizon mirror is inclined 80° to the axis of the telescope, or that the angle intercepted between the ray incident on the horizon glass and the corresponding reflected ray passing through the telescope is 200, which is the case in circular instruments of DE BORDA'S Construction, and on this supposition the errors of an instrument in measuring different angles may be ascertained by the rules in pages 98 and 106; when the intercepted angle is greater or less than 200, which is the case in most sextants and quadrants, the error in any measured angle corresponding to an inclination of the surfaces of 1', may be obtained as follows. Find in the first column the intercepted angle, and the sum of that angle and the observed distance; take the corresponding corrections from column 5th, and their difference will be the sought correction. In a circular instrument you must find in the side column the sum and the difference of the intercepted angle and observed angle, and take out the corresponding corrections from column 5th, half their difference will be the sought correction. Having thus found the correction corresponding to 1', you may find the correction for other angles as in pages 98 and 106. TABLE XXXV. Correction for a deviation of the telescope of an instrument of reflection from the parallelism to the plane of the instrument. The uses of this table are explained in pages 97 and 105. TABLE XXXVI. Correction of the mean refraction for various heights of the barometer and thermometer.-The use of this table is explained in page 108. TABLE XXXVII. Latitudes and Longitudes of the fixed Stars.-This table contains the Latitudes and Longitudes of the principal fixed stars, adapted to the beginning of the year 1820, with the annual variations for precession and the secular equation, by which the mean values at any time may be obtained, in like manner as the Right Ascensions and Declinations are from Table VIII.; by adding the correction of longitude after 1820, subtracting before 1820, and applying the correction of latitude with the same sign as in the table after 1820, but with a contrary sign |