ing 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 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 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 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 I. EXAMPLE II. Required the degrees and minutes corresponding to 6h. 33m. 20s. Opposite 6h. 32m. Os.......in col. 4 is.. 1 20 .in col. 2 is....... 6 33 20 .... 98° 0 20 98 20 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 is 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.; and 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 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: EXAMPLE I. : 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 with the degrees, and at the side marked M. with the minutes: the corresponding numbers 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 this 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 angle; 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 25° 42′ 19′′. In like manner, we find the cosine of the same angle to be 90108-490104, 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. TÁBLE 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 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. If the declination had been decreasing, the horary motion would be subtractive instead of additive, as m. 8. 2.9 .1 Right Ascension, 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 11d. Gives 's Rt. Asc. May 11d. EXAMPLE VI. 152.17-112.17 106 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. 2870.14 47.1 Sub. from declination Sept. 10d. 8h. 9 32 13.3 EXAMPLE VII. Required the moon's declination in 1836, May 11d. 17h. 35m. 3os. mean time, astronomical account, at Greenwich. Here the motion in declination for 10m. is by N. A. 143".02. Motion for 30m. is 143".02 × 3 = ........ 429".1 Tab. XXX. 143" at top, and 36s. at side in col. M. the corr. divided by 10 is 8.6 509.2 Add to declination May 11d. 17h. by N. A.... 8′ 29′′.2 2 19 25.9 Motion in declination is D's declination May 11d. 17h. 35m. 36s..... Here the correction 8 29.2 is added, because the declination is increasing. 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 when a star comes to the meridian. EXAMPLE II. At what time was Pollux on the meridian of a place in the longitude of 70° 46′ W., March 31, 1836, sea account? March 31, sea account, is March 30, N. A., on which day, at noon, the sun's right h. m. s. Gives the approximate time of southing... 6 59 11 0 37 53 Right ascension of Pollux................ 7 35 17 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 sub tracted in the afternoon, and we shall have the corrected 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 than 12h., it will be after midnight; but if less than 12h., it will be before midnight. 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 the examples of finding the latitude by one altitude taken near noon, given in the body of the work, pages 201–203. 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 20°, which is the case in circular instruments of DE BORDA's construction, and on this supposition the errors of an instrument in measur ing different angles may be ascertained by the rules in pages 136 and 143; when the intercepted angle is greater or less than 20°, 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 136 and 143. 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 135, and 143. TABLE XXXVI. Correction of the Mean Refraction for Various Heights of the Barometer and Thermometer.-The use of this table is explained in page 154. 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 1830, 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 1830, subtracting before 1830, and applying the correction of latitude with the same sign as in the table after 1830, but with a contrary sign before 1830. The latitudes and longitudes, thus obtained, are the mean values. When great accuracy is required, the corrections for the equation of the equinoxes, Table XL. and aberration, Table XLI. must be applied. TABLE XXXVIII. Reduction of Latitude and Horizontal Parallax.-This table contains the corrections to be subtracted from the latitude of the place of observation, and from the horizontal parallax of the moon, given in the Nautical Almanac, in calculating eclipses of the sun or occultations. Thus, if the latitude of the place was 40°, and the moon's horizontal parallax 57', the correction of latitude would be nearly -I 18", and that of parallax-4.7, so that the reduced latitude would be 39° 48′ 42′′, and the reduced parallax 56 55.3. These values are to be used in occultations; but in eclipses of the sun, this parallax is to be further decreased by 8.6 for the sun's parallax. When the latitude is not given exactly in the table, the two nearest numbers must be found, and a proportional part of their difference is to be applied to one of the numbers, as usual. In calculating this table, the ellipticity of the earth was supposed equal to 3, as in the third edition of La Lande's Astronomy, and in Vince's Astronomy. This value differs but little from 307.6 and 305.95, deduced by La Place from two lunar equations in the third volume of his immortal work, La Mécanique Céléste. In the second volume of the same work, he calculated the ellipticity to be from the lengths of pendulums observed in different lati tenth of his equations A" becomes 15, which does not differ very much from the value assumed in this table. TABLE XXXIX. Aberration of the Planets.-This table contains the aberration of the planets, to be applied to the true longitude or latitude, with the same sign as in the table. The argument at the side is the elongation of the planet from the sun; that is, the difference of their geocentric longitudes, or its supplement to 360°. Thus, on July 19, 1820, the longitude of the sun was 3s. 26° 38', the geo. long. of Venus 4s. 13 23, their difference 16° 45' is the elongation or distance from the inferior conjunction, corresponding to which is the aberration 3" to be applied to the true longitude given by the tables to obtain the apparent longitude. The aberration of Mercury is given at its greatest, least and mean distances from the sun. At the intermediate places, a proportional part of the differences of the nearest tabular numbers must be applied. TABLES XL. and XLI. Equation of the Equinoxes and Aberration in Longitude.— Table XL. contains the equation of the equinoxes in longitude common to all the heavenly bodies. The argument is the longitude of the moon's ascending node; the signs of longitude being found at the top or bottom, and the degrees at the side, the corresponding number with its sign is the equation of the equinoxes in longitude. Table XLI. contains the aberration of the stars in longitude and latitude, to be calculated by the rules at the bottom of the tables; the signs of the argument being found at the top, and the degrees at the side,* taking proportional parts for minutes. The corrections of longitude found in these tables are to be applied, with their signs, to the mean longitude found in Table XXXVII., and the correction of latitude, Table XLI., is to be applied to the mean latitude deduced from Table XXXVII. Thus, on July 16, 1830, by the examples at the bottom of Tables XL. XLI., the equation of the equinoxes was-5.3, and the aberration in longitude 11.3; these corrections being applied to the mean longitude of the star deduced from Table XXXVII., 11s. 21° 7 327, gives its apparent longitude 11s. 21° 7′ 38′′. In a similar manner the aberration in latitude, -5.6, found at the bottom of Table XLI., applied to the mean latitude, 19° 24′ 45′′ N., deduced from Table XXXVII., gives the apparent latitude of the star 19° 24′ 39′′ N. TABLES XLII. XLIII. Aberration and Nutation in Right Ascension and Declination.— Table XLII. contains the aberration, and Table XLIII. the nutation in right ascension and declination, to be found by the rules at the bottom of the tables, and applied, with their signs, to the mean values deduced from Table VIII. Thus, by Table VIII., the right ascension of a Pegasi, July 16, 1830, was 22h. 56m. 20s., and its declination 14 15 N. The aberration of right ascension in time was nearly+0s.8, in declination — 0.8; the nutation in right ascension in time-Os.1, in declination +0.5, as appears by the examples at the bottom of the tables. These corrections being applied to the mean values, give the apparent right ascension 22h. 56m. 21s., and the apparent declination 14° 18′ N. The equation of the obliquity of the ecliptic may be calculated by the rule at the bottom of the table. Thus, on July 16, 1830, the equation was - 9.1, which, applied to the mean obliquity 23° 27 42.0, gives the apparent obliquity 23° 27′ 32.9. TABLE XLIV. Augmentation of the Moon's Semi-diameter.—This table is divided into four parts, and is useful in finding the augmentation of the moon's semi-diameter by means of the altitude and longitude of the nonagesimal when the moon's altitude is unknown. The precepts for this calculation are given at the bottom of the table, and for further illustration another example is added, in which it is required to find the augmentation at the commencement of the occultation calculated in Problem VII. of the Appendix, when the D's S. D. by the Nautical Almanac was 16' 18.9, her true latitude 1° 55' 11" S., parallax in lat. 10' 23.6, altitude of the nonagesimal 81° 17′ 32", and the moon's apparent distance from the nonagesimal 51° 38' 26", as in Example III. Prob. V. Appendix In this case the arguments of Part I. are 81° 17' 32" + 51° 38′ 26", or nearly 4s. 12° 56' and Os 29° 39′, and the corresponding corrections +6.00, +4.05, whose sum is 10.05. This in Part II. gives +0.10. In Part III., with the moon's true latitude, 1° 55' 11" S., and her par. in lat. 10 23.6, the correction is 0.10. The sum of these three parts is +10.05, which being found at the side of Part IV., and the moon's horizontal S. D. 16' 18.9 at the top, gives the corresponding correction +0.40. This connected with the three former parts +10.05, gives the sought augmentation 10.45, or 10.4, as in the example Prob. VII. Appendix. It may be observed that the calculation by Problem IV. will sometimes produce the supplement of the altitude of the nonagesimal; but this requires no alteration in the rule, since the result is the same whether the altitude or its supplement is used. TABLE XLV. Equation of Second Differences.-This table contains the equation of the second differences of the moon's motion, or the correction to be made on account of her unequal velocity between the times marked in the Nautical Almanac. The manner of applying this correction is taught in Problems I. II. III. of the Appendix. TABLE XLVI. Variation of the Altitude of an Object, arising from a Change of 100 Seconds in the Declination.-This table is useful in finding the latitude by double altitudes of the sun, or any other object. It is explained in the precepts for such calculations, pages 189, 190, 191, &e. The table is to be entered at the top with the latitude of the place, and *The degrees in this and the following tables are to be found in the column marked D on the same horizontal line with the signs. Thus if the signs are at the top of the table, the degrees must be found in the left colums, otherwise in the right. |