and the point n the place of noon or 0 hours. If it had been south, the point I would have been marked Oh., and the points marked XI, X, &c. would be I, II, &c. respectively. The path touches the circle ARB in two points, representing the points of sun rising and setting, which, in the present figure, are respectively 16h. 26m. and 7h. 34m. These points divide the path into two parts, of which one represents the path by day, the other by night, as is evident from the hours marked on the curve. Half hours, or any other intermediate time, may be marked in a similar manner. Thus, for the time 3h. 30m.=52° 30', set the sine of 52 to the radius r VI, from r to h on the liner VI, and erect the perpendicular hi equal to the sine of 370 (which is the complement of 52°) to the radius rn, and the point i will be the place of the spectator at the proposed time. In this way the halves and quarters of hours may be marked on those parts of the path where necessary. The smaller subdivisions may generally be obtained to a sufficient degree of accuracy by dividing the quarters of hours into equal parts. Take from the scale of equal parts an extent equal to the sum of the semi-diameters of the sun and moon, and, beginning near N, find, by trials, the point p' of the moon's path, and the point Z' of the path of the spectator, marked with the same time and at that distance apart. That time will be the beginning of the eclipse. If no such points can be found, there will be no eclipse at the proposed place. Proceed in the same way towards the point L, and find the points p", Z", at the same distance apart; the corresponding time will be the end of the eclipse. Find, by trials, the point p of the moon's path, and the point Z of the path of the spectator, marked with the same times at the nearest distance from each other (which will in general be nearly the middle time between the beginning and end of the eclipse); that time will be the middle of the eclipse. On Z as a centre, with a radius equal to the sun's semi-diameter, describe the circle whose diameter is Ss, representing the sun's disc; and on the centre p, with a radius equal to the moon's semi-diameter, describe the circle whose diameter is Mm, representing the moon's disc. The part of the sun's disc that is cut off by this circle will represent the part of the sun that is eclipsed. In the example of figure 10, the centre, p, of the moon's disc is so near that of the sun, Z, that the eclipse is nearly central; and, as the moon's semi-diameter is greater than the sun's, the eclipse must be total. Under similar circumstances, if the moon's semi-diameter had been least, the eclipse could have been annular. In case of a partial eclipse, the sun's disc will not be wholly covered by the moon, as in figure 11, Plate XIII., where the circles representing the discs of the sun and moon are marked with the same letters as in figure 10, but the objects are placed in a different situation. In this case, the number of digits eclipsed may be obtained by drawing a line through the centres p, Z, to meet the discs in the points S, M, s, m, and by saying, As the distance Ss (representing the whole disc) is to the obscured point Ms, so are 12 digits to the number of digits eclipsed. The beginning and end of total darkness in a total eclipse are found like the beginning and end of the eclipse, except in taking in the compasses the difference between the semi-diameters of the sun and moon, instead of their sum. For the points of the path of the spectator and of the moon's orbit, marked with the same time, and at that distance from each other, will represent the situations and times of the beginning and end of total darkness. The beginning and end of the internal contacts of an annular eclipse are found in the same manner; the only difference is that, in a total eclipse, the moon's semi-diameter is greatest, but in an annular eclipse the least. In observing the beginning of a solar eclipse, it is of some importance for the accuracy of the observation, to know on what part of the sun's limb the eclipse will begin. This is easily found by means of the projection. Thus at the beginning of the eclipse, which corresponds to the point p' of the moon's path, and the point Z' of the path of the spectator, the first point of contact g may be obtained by drawing about the centre p', with a radius equal to the moon's semi-diameter, a circle representing the moon's disc;* about Z' as a centre, with a radius equal to the sun's semi-diameter, another circle representing the sun's disc, touching the former in the point g. Draw the line CZ', meeting the sun's disc in the points a, c, the point c being the most distant from the centre C. Then the circle g a c, being held between the eye of the observer and the sun, the engraved or marked side of the figure towards the eye, and the line ca in a vertical direction with the point c uppermost, will represent the appearance of the sun as viewed by the naked eye at that time; c will represent the upper part of the sun, a the lower, and g the point of contact. If the eclipse be observed with an inverting telescope, the contrary will be observed; that is, the part a must be uppermost, c the lowest, and g, the point of contact, will appear to the left hand of ca. In a similar manner the appearance of the objects may be obtained at any other part of the eclipse, but it is not necessary except at the beginning of it, where there is nothing to direct the eye of the observer. * Instead of this circle, the line p' Z may be drawn cutting the sun's disc in the sought point of contact g EXAMPLE. Required the times and phases of the total eclipse of the sun, June 16, 1806, at Salem, in the latitude of 42° 33′ 30′′ N., and the longitude of 4h. 43m. 32s. west from Greenwich. By the Nautical Almanac, the time of new moon at Greenwich was June 16d, 4h. 19m., corresponding to June 15, 23h. 35m. 28s., at Salem. At the time at Greenwich, 4h. 19m. the elements of the eclipse were, as in the adjoined table, calculated by the above rule. ELEMENTS. Conjunction at Greenwich, June 16,... Sum of semi-diameters......... D's horary motion in longitude, Prob. II. h. m. s. 4 19 00 4 43 32 23 35 2 42° 33′ 30 60 25.7 8.6 60 17.1 16 28.1 15 46.1 32 14,2 42.0 36 41.2 2 23 1 ..........CO 34 18.1 .DF 23 22 N. Draw ACB (Plate XIII. fig. 10), and perpendicular thereto the line CGR. Make CG equal to the moon's latitude, 19′ 37′′ N., taken from a scale of equal parts, the point G being above C because the latitude is north. Make CO equal to the moon's horary motion from the sun, 34' 18.1, to the right hand of the point C; and CP equal to the moon's horary motion in latitude 3′ 22.5, the point P being below C because this horary motion has the sign+prefixed. Draw NGL parallel to OP. Make OP a transverse distance of 60, 60, on the line of lines of the sector, and measure from the same lines the transverse distance 35, 354 (corresponding nearly to the minutes in the time of new moon); this distance, set on the line GN to the right of G, reaches the point x, where the hour preceding the new moon is to be marked, viz. 23h. Take OP in the compasses, and mark it successively on the line NL from x, or 23h., to the right to 22h., and to the left to 24h. or Oh., 1h., &c. These are subdivided into five minutes, the scale not admitting smaller divisions. Take the moon's reduced horizontal parallax, 60' 17.1, from the scale of equal parts, and with that radius describe about the centre C the circle ARB. Set off (by means of the sector) the arcs RT, RU, each equal to 23° 28'. Join TQU, and about that diameter describe the circle TYU. Make the arc TV equal to the sun's longitude, 84° 44' 36", which is done by setting the radius QT as a chord from T to I, and then the arc V= 24° 44' 36" by means of the sector. Draw P/V parallel to CR, to meet TU in the point P'. Join CP', and continue it to meet the circle ARB in W. Make (by the sector) the arcs WD, Wd, equal to the complement of the latitude of the place, 47° 26' nearly, the radius being CB. În a similar manner make the arcs DF, DE, df, de, &c., each equal to the sun's declination 23° 22. Draw the lines Flf, Dqd, Ene, cutting CW in 1, q, n. Bisect in in r. Draw the line VI XVIII parallel to Dqd, and make r VI, r XVIII, each equal to qD. Through the points 1, VI, n, XVIII, l, draw the path of the spectator as taught in the above rule, and mark the hour of noon, Oh., at the point n because the sun's declination is north. Mark the following hours in succession to the left, I, II, III, &c., as in the figure. Take an extent in the compasses equal to the sum of the semi-diameters of the sun and moon, 32′ 14′′.2, and, beginning towards N, find, as above directed, the points p'Z' at that distance apart and marked with the same time, 22h. 7m. nearly, which is the time of the beginning of the eclipse. Proceed in the same way for the end of the eclipse corresponding to the points p', Z", and to the time Oh. 53m., which is the time of the end of the eclipse. Take the difference of the semi-diameters of the sun and moon, 42", in the compasses, and proceed in the same way to find the beginning and end of total darkness, 23h. 27m., and 23h. 31m. The points corresponding could not be drawn in the figure, as they are so near to p and Z, and the scale smail. Find, by trials, the points p, Z, marked with the same time and at the least distance apart; this will be the time of the middle of the eclipse, 23h. 29m. With an extent equal to the moon's semi-diameter, 16' 28.1, as a radius, describe about p the circle whose diameter is Mm representing the moon's disc; and with the sun's semidiameter, 15' 46.1, describe about Z the circle whose diameter is Ss, representing the sun's disc at the middle of the eclipse. The sun's disc being wholly covered by the moon, indicates that the eclipse was total. Describe, in the same way, about p' and Z' the discs of the sun and moon, at the beginning of the eclipse, touching each other in g. Draw CZ', cut ting the moon's disc in c and a. Then the arc cg will be the distance of the first point of contact of the sun and moon from the sun's zenith towards the western part of the limb. REMARKS. 1. The correction for the spheroidal form of the earth. the augmentation of the moon's semi-diameter, inflexion and irradiation, are neglected in the above rule, as not sensibly affecting the result of the projection, though these points might be attended to by the following precepts. 2. From the latitude of the place subtract the correction of latitude of Table XXXVIII., and from the moon's horizontal parallax, decreased by 8.6, subtract the correction of paral lax in the same table; the remainders will be the corrected latitude and parallax to be made 3. Decrease the moon's semi-diameter given by the Nautical Almanac by 2" for inflexion, if it be thought necessary. 4. Decrease the sun's semi-diameter 33" for irradiation, and from the remainder subtract a correction equal to the augmentation (Table XV.) that the moon's semi-diameter would have when at the same altitude as the sun; the remainder will be the corrected semi-diameter of the sun, to be used in the above rule in finding all the times and phases of the eclipse. This method of decreasing the sun's semi-diameter produces nearly the same result as that by augmenting the moon's semi-diameter, horary motion, and horizontal parallax, and taking the sun's semi-diameter as given in the Nautical Almanac. 5. Besides these corrections, there are others, depending on the change of the moon's semi-diameter, horizontal parallax, and horary motion during the eclipse; but all these corrections are usually neglected in projecting an eclipse or occultation. 6. The altitude of the sun, which is nearly the same as that of the moon during the eclipse, may easily be found by means of the projection. Thus, if it were required at the beginning of the eclipse, when the spectator is at Z: Take the distance CB, and apply it as a transverse distance 90°, 90°, to the sines of the sector; then the distance CZ', applied in the same manner to those lines, will give the zenith distance of the sun, about 31°, corresponding to the altitude 59°. The correction (Table XV.) corresponding to this altitude is 14", which is nearly the correction to be subtracted from the sun's semi-diameter, 15/42.6 (corrected for irradiation), to obtain the corrected semi-diameter, 15' 28.6, as taught in §4. Table XV. was calculated for the mean semi-diameter, 15' 37", and the correction of the Table, 14", ought to be increased in ratio of the sun's semi-diameter, 15' 46.1, to 15' 37", when very great accuracy is required. The difference of the corrected semi-diameters of the sun and moon, 15' 28.6 and 16' 26.1, is 57", which is to be used instead of 42" in finding the beginning and end of total darkness. The duration of the total darkness found by the corrected value 57", is 4 minutes, but with the uncorrected value 42", is only 3 minutes. It was probably owing to the neglect of this correction that some of the Almanacs published in this country, for 1806, mentioned the duration as 3 minutes. 7. The path of the spectator, I, II, III, IV, &c., calculated for the proposed latitude 42° 33′ 30′′, may be made to answer for any other latitude by altering the centre of projection and the scale of equal parts. By this means the trouble of repeatedly describing that path, when the eclipse is to be calculated for several places, may be avoided. To do this, add. the prop. log. of the reduced parallax to the log. secant of the latitude of the place; the sum, rejecting 10 in the index, will be the prop. log. of an arc A. To this prop. log. add the log. secant of the sun's declination (or star's in an occultation), and the log. cotangent of the latitude of the place; the sum, rejecting 20 in the index, will be the prop. log, of the arc B. Take the radius r VI (or qD), in the compasses, and make it a transverse distance on the line of lines of the sector corresponding to the arc A, and with that opening of the sector measure the transverse distance corresponding to the arc B, which, set from r towards C on the line rC (continued if necessary), will reach to the centre of the projection corresponding to the proposed latitude; the transverse distance corresponding to the reduced parallax, measured from the line of lines with the same opening, will be the radius of the projection, and the transverse distance corresponding to the horary motion of the moon from the sun or star, in an occultation, will be the horary distance to be made use of in marking the hours on the lunar orbit LN; lastly, the latitude of the moon at the conjunction is to be measured as a transverse distance, and set from the new centre of projection on a line drawn through it parallel to CR, and the point where it reaches will be the new point G, corresponding to the place of the moon at the ecliptic conjunction. Through this point the line of the moon's path is to be drawn parallel to the line LN of the figure, and the hours are to be marked on it as before. Whence the times of beginning and end of the eclipse may be found as in the above rule. An example of this method is not given, as it would render the scheme too confused. PROBLEM XII. To project an occultation of a fixed star by the moon, at any given place. The method of projecting an occultation is nearly the same as that of an eclipse of the sun; but to save the trouble of reference, it was thought expedient to give the rule without abridg ment. RULE. To the time of the ecliptic conjunction of the moon and star, computed from the Nautical Almanac by Problem III., add the longitude of the proposed place turned into time, if east, but subtract if west; the sum or difference will be the time of conjunction at the proposed place. Corresponding to the time of conjunction at Greenwich, find, by Problem I., the moon's latitude, horizontal parallax, and semi-diameter; also the sun's right ascension. Then, by Problem II., find the horary motion of the moon in longitude and latitude, and by Tables VIII. and XXXVII., the star's right ascension, declination, longitude and latitude.* * In strictness, these quantities ought to be corrected for aberration and nutation, by Tables XXXIX. XLIII., but the correction is so small that it may always be neglected. If the right ascension and declination only are given, the latitude and longitude may be found by Problem XIX., and if the latter are given, the former may be calculated by Problem XX. It will be found most convenient to use the right ascensions and declinations which are given in the Nautical Almanac, when any of the stars marked in it are used. Draw the line ACB (Plate XIII. fig. 8), representing a parallel of the ecliptic passing through the star, and perpendicular thereto the line CPR. Take a scale of equal parts to measure the lines of projection, and from it take an interval equal to the difference of the latitudes of the moon and star, and apply it to the line CR from C to G, above the line ACB if the moon's latitude is north of the star's, otherwise below.* Take CO equal to the horary motion of the moon in longitude, and set it on the line CB to the right hand of C to O; take CP equal to the moon's horary motion in latitude, found with its sign by Problem II., and set it on the line CR from C to P, above the line ACB if its sign is, below if + Join OP, which represents the horary motion of the moon in her orbit, and parallel to that line draw the orbit of the moon, NGL, on which are to be marked the places of the moon before and after the conjunction by means of the horary motion OP, so that the moment of the ecliptic conjunction at the proposed place may fall exactly at the point G, as in the figure where the conjunction is at 18h. 42m. This may be done by making OP equal to the transverse distance 60, 60, on the line of lines of the sector, then measuring from the same lines the transverse distance corresponding to the minutes and parts of a minute in the time of the ecliptic conjunction at the place of observation, and setting it on the line GN from G towards the right to the point x, the place of the moon at the first whole hourt preceding the conjunction (which in the present figure is 18h.) Then the distance OP, being taken in the compasses, and set from z to the right hand, gives successively the preceding hours, and the same distance set to the left gives the following hours, as in the figure, where they are marked 17h., 18h., 19h., 20h. These hours are to be divided into 60 equal parts representing minutes, the scale being taken sufficiently large for that purpose.§ In the present figure the subdivisions are carried only to five minutes. Take the moon's horizontal parallax from the scale of equal parts for the radius CB, with which, on the centre C, describe the circle BRA, cutting CR in R. Open the sector till the transverse distance 60°, 60°, on the line of chords is equal to the radius CB, and measure from that line the transverse distance 23° 29′ (equal to the obliquity of the ecliptic), which set on the circle ARB, on each side of R to T and U. Join TU cutting CR in Q. On Q as a centre, with the radius QT, describe a circle, TYUV, on which set off the arc TYV, equal to the star's longitude. Through V draw the line VP' parallel to CR. Open the sector till the transverse distance 90°, 90°, on the sines, is equal to the radius CB; then take in the compasses from the same lines an extent equal to the transverse distance corresponding to the complement of the declination of the star, and with one foot in C sweep a small arc to cut the line VP' in P', the place of the pole of the earth.|| Draw CP', and continue it on either side so as to cut the circle ARB in the point W, situated above AB, if the latitude of the proposed place is north, but below if south. In the proposed figure the latitude is north. (If it had been south, the lower part of the circle ARB ought to have been made use of.) Open the sector as before, so as to make the transverse distance of 60°, 60°, on the chords, equal to CB, and take the chord of the complement of the latitude of the place, which set from W on each side to D and d. With the same opening of the sector measure the chord of the star's declination, which set on the circle ARB from the point D on each side, to E and F, and from d on each side to e and f. Draw the dotted lines Ff, Dd, Ee, cutting CW in l, q, n. Bisect In in r, and erect the line tru perpendicular to CW, and make rt, ru, each equal to qD. Open the sector to make the transverse distance 90°, 90°, on the sines equal to rt, and on each side of r mark on the line tru the sines of 150, 30°, 45°, 60°, 75° (equal to 1h., 2h., 3h., 4h., 5h, respectively) to that radius, and mark the points with those degrees as in the figure; through these points draw the dotted lines parallel to In as in the figure. Open the sector so that the radius r may correspond to the transverse distance 90°, 90°, on the sines, and measure the complements of the former degrees as transverse distances on the sines, viz. 75°, 60°, 45°, 30°, 15°, and set them on the above dotted lines, on each side of the points 15°, 30°, &c., respectively, above and below the line tru. A regular curve, ntlun, drawn through the extremities of these dotted lines, will represent the path of the spectator in the given latitude. Subtract the sun's right ascension from the star's (increasing the latter by 24 hours when necessary); the remainder will be the hour of the star's passing the meridian,¶ which is to be marked at the upper point I of the path if the star's declination is south, but at the lower point n if the declination is north. The other hours are to be marked from this point towards the left, by marking successively, at the points where the dotted lines meet the path, the hour of the star's passing the meridian, increased by 1h., 2h., 3h., &c., completely round the curve, observing to reject 24 hours when the sum exceeds 24h. In the present example, the star's declination is south; consequently the upper point of the path is taken for the hour of passing the meridian, 19h. 54m.; the extremities of the dotted lines to the left being marked successively 20h. 54m., 21h. 54m., 22h. 54m., 23h. 54m., Oh. 54m., &e. In the figure the point G is placed above ACB, because the moon is in a less southern latitude than the star. This part of the rule may also be thus expressed :-Find the moon's latitude with its sign as in Prob lem II. Prefix the sign + to the star's latitude if north, the sign if south. Add the latitudes, noticing the signs as in algebra, and the distance CG will be obtained. If its sign is, the point G is to be placed above C, but below C if the sign is +. † See note with the mark in page 416. See note with the mark fin page 416. See note with the mark in page 418. The distance of the line WV from the line CR, the situation of the point P', and the path of the spectator, may be found as in the note page 418. Or rather the horary distance of the sun and star at the time of the ecliptic conjunction of the moon . The path touches the circle ARB in two points, representing the points of rising and setting of the star, which, in the present figure, are 14h. 9m., and 1h. 39m. These points divide the path into two parts, of which one represents the path while the star is above the horizon, the other when below, as is evident from the hours marked on the curve. The half hours, or any other intermediate time, may be marked in a similar manner. Thus, for the time 4h. 24m., which is 3h. 30m., or 52° 30', from the time 7h. 54m., marked at the point n, set the sine of 52° to the radius rt from r to h on the line rt, and erect the perpendicular hi, equal to the sine of 3740 (which is the complement of 5230), to the radius rn, and the point i will represent the place of the spectator at the proposed time. In this way the halves and quarters of hours may be marked on those parts of the path where necessary. The smaller subdivisions may generally be obtained to a sufficient degree of exactness by dividing the quarters of hours into equal parts. Take from the scale of equal parts an extent equal to the semi-diameter of the moon, and, beginning at the line NL, towards N, find, by trials, the point p' of the moon's path, and the point Z' of the path of the spectator, marked with the same time and at that distance apart. That time will be the beginning of the occultation or immersion at the proposed place. Proceed in the same way towards the point L, and find the points p, Z, at the same distance apart; the corresponding time will be the end of the occultation or emersion. About the points p', p, as centres, with a radius equal to the moon's semi-diameter, describe the small circles meeting the paths of the spectator in the points Z', Z. These circles will represent the moon's disc; the points Z', Z, the places of the star, and the line CZ', CZ, the vertical circles passing through the star at the times of immersion and emersion respectively. To render this part of the scheme more distinct to the eye, it is drawn separately in figure 9, Plate XIII., in which the points C, p',Z', are similarly situated to the corresponding points of figure 8, marked with the same letters. Through p/ draw the line a' p'c' parallel to CZ, to meet the moon's disc in a', c'. Then the circle a' Z' c', being held between the eye of the observer and the sun, the engraved or marked side of the figure towards the eye, and the line CZ' (or a p' c') in a vertical position with the point Z above C, will represent the appearance of the moon and star as viewed by the naked eye; c' will represent the upper part of the moon, a' the lower part, and Z' the point of contact. The contrary will be observed if the object be viewed by an inverting telescope. It will generally be conducive to the accuracy of an observation, to estimate in this manner the point of emersion, so as to keep that point of the moon's limb in the field of view of the telescope, and the eye directed towards that point of the limb, so as to perceive the star at the first instant of its appearance. The situation of the point of emersion with respect to the horns g, 6, of the moon may also be made use of for this purpose. The line ope, connecting the moon's horns, is nearly parallel to the line CR, except very near the new or full moon; so that in general it will be sufficiently correct to draw through p the line e pe parallel to CR. If greater accuracy is required, the following construction may be made use of. Subtract the sun's longitude from the moon's,t make the arc TYUA equal to the remainder, and join Q. Set on the same circle the arc T equal to the moon's latitude; below the point T if that latitude is south, above if north. Through ẞ draw the lined parallel to TQ to cut Q in and CR in d. Take the extent QT and set it on the line dY above 8 to μ. Join ue, and parallel thereto through p draw the line ope cutting the moon's disc in the points o representing the horns, the figure being viewed as above directed. The enlightened part of the moon is that nearest to the sun; the dark part is the most distant from it. EXAMPLE. Required the times of immersion and emersion of Spica, December 12, 1808, at a place in the latitude of 20° N., and in the longitude of 1h. 9m. east from Greenwich. By the first page of the Nautical Almanac for the month of December, 1808, the time of the ecliptic conjunction of the moon and Spica (marked Da I) was December 12, 17h. 33m. at Greenwich, corresponding to 18h. 42m. at the proposed place. This time may also be computed by means of the longitudes of the objects, as in Problem III. of this Appendix. At the time at Greenwich, 17h. 33m., the elements of the occultation were, as in the adjoined table, calculated by the above rule. Draw ACB, and perpendicular thereto the line CGY. Make CG equal to the difference between the In strictness, the longitude and latitude of the moon at the time of immersion or emersion ought to be made use of; but it will be sufficiently exact to use the star's longitude instead of the moon's (increasing it by 360° when less than the sun's longitude), and the moon's latitude at the conjunction. Quantities of the <ine order as the moon's parallax are neglected in the value of the arc TYUX. |