representing the moon's disc, and with the sun's semi-diameter 15′ 46′′.1, describe about Z the circle whose diameter is S s representing the sun's disc at the middle of the eclipse. The sun's dise 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' cutting the moon's dise in c and a. Then the arch c g 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 Tab. XXXVIII. and from the moon's horizontal parallax decreased by 8'.8 subtract the correction of parallax in the same table; the remainders will be the corrected latitude and parallax to be made use of in the above rule to correct for the spheroidal form of the earth. 3. Decrease the moon's semi-diameter given by the N. A. by 2" for inflexion. 4. Decrease the sun's semi-diameter 3' for irradiation, and from the remainder subfract a correction equal to the augmentation (Tab. 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 semidiameter 15', 42".6 (corrected for irradiation) to obtain the corrected semi-diameter 15′ 29′′.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 semidiameter 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 total rkness. The duration of the total darkness found by the corrected values 57 is 43 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 arch A. To this Prop. Log. add the log. secant of the sun's declination (or star's in an occultation) and the log. co-tangent of the latitude of the place, the sum, rejecting 20 in the index, will be the Prop. Log. of the arch 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 arch A, and with that opening of the sector measure the transverse distance corresponding to the arch 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 EXAMPLE. Suppose in a place in the latitude of 20° 0' N. longitude 11. 9m. Os. east of Greenwi the occultation of Spica by the Moon on Dec. 12, 1808, was observed; the immersion at 16 sion at 18h. 10' 29'', apparent time by astronomical computation. Required the longitud observation? Most of the elements in the following Table are calculated by Problems I. II. and VI. | IMMERSION d. h. 12 15 48 29 E -Reduction, Table XXXVIII. Reduced parallax D's S. Diam, by N. A.-Indexion 2' Add correction Table XLIV. 59 52.3 1.4 59 509 16 16.9 10.4 16 27.3 S1 $5 51.7 46 25 200 54 21.3 D's true lat. by N. A. Prob. 1. D's parallax in latitude D's apparent latitude South Difference D app. lat. Difference of D's app. latitudes 's tr. lat. lat. T. XXXVII. 2° 2' 13.9. S.-T. XLI. 06 3+Tab. 's tr. long. Long Tab. XXXVII. 201° 10′ 29" 3+1 The difference of the apparent latitudes of the Moon and Star at the beginning of The moon's horary motion varies from 35 517 to 35' 54".2 during the occultation middle time 17h. 49′ 45′′ between the immersion 16h, 57' 29" and the conjunction 12 from the Nautical Almanac) the horary motion was 35' 53".5 as is easily found by a cal to that in the Example of Pro The difference of meridians deduced from the observation 1h. 9' 2" differs but 2′′ from the assumed quantity 1h. 9′ 0′′. If the difference had been considerable, it would have been necessary to repeat the operation with the difference of meridians thus calculated, and so on till the assumed and calculated longitudes agree. The errors of the tables above found were deduced upon the supposition that the observations were actually made at the place mentioned in this example, and that the true longitude of the place of observation was 1h. 9 0". For it must be observed, that the errors of the tables in longitude cannot be found by an observation of an eclipse or occultation without knowing by other observations the precise longitude of the place of observation. This is evident by observing, that by repeating the operation till the assumed and calculated longitude of the place of observation agree with each other, the long. of the moon, deduced from the calculation, will agree also with the longitude by the tables. The time of conjunction at Greenwich 17h. 33′ 0′′ taken from the Nautical Almanac, is liable to a small error from the incorrectness of the tables. To obviate this error it will be necessary to deduce (by the above method or by Problem IX. when only the beginning or end is observed) the time of conjunction from observations actually made at two places, the difference of these times will be the difference of meridians free from the errors of the tables. PROBLEM VIII. To find the longitude of a place by an eclipse of the sun when the beginning or end only is observed, the apparent time being estimated from noon to noon, according to the method of astronomers; the latitude of the place being also known. RULE. To the apparent time apply the estimated longitude of the place in time, by adding if west, subtract ing if east, the sum or difference will be the supposed ticae at Greenwich. Corresponding to this time in the Nautical Almanac, find by Problem I, the moon's semi-diameter, horizontal parallax, longitude, and latitude; and the sun's semi-diameter, longitude, and right ascension; also the moon's horary motion from the sin by Problem H. Decrease the sun's semi-diameter 3 for irradiation. Decrease the moon's semi-diameter 2' for inflexion, and to the remainder add the correction to Table XLIV. † the sum will be the moon's corrected semi-diameter. Find also, in the Nautical Almanac, the obliquity of the ecliptic. With these elements and the apparent time at the place of observation, calculate the altitude and longitude of the nonagesimal by Problem IV. and the parallaxes in longitude and latitude and the moon's apparent latitude by Problem V. To the sum of the corrected semi-diameters of the sun and moon add and subtract the moon's apparent latitude, and find the logarithms of the sum and difference in seconds. Half the sum of these two loga. richins will be the logarithm of an arch in seconds, to be added to the sun's longitude if the phase is after the apparent conjunction, but subtracted if before: the sum or diference will be the apparent longitude of the moon. To this add the moon's parallax in longitude when the moon's distance from the nonagesimal (found as in Problem VI. by subtracting the longitude of the nonagesimal from the moon's longitude, borrowing 360 when necessary) is greater than 160, otherwise subtracted, the sum or difference will be the true longitude of the moon. Take the difference in seconds between the true longitudes of the sun and moon, and to its logarithm add the arithmetical complement log. of the moon's horary motion from the sun in seconds, and the constant logarithm 3.55650, the sum, rejecting 10 in the index, will be the logarithm of the correction of the given time expressed in seconds. This is to be added to the apparent time of observation when the moon's true longitude is less than the sun's, otherwise subtracted; the sum or difference will be the time of the true conjunction at the place of observation. The difference between this and the time of conjunction inferred from the Nautical Almanac for the meridian of Greenwich by Problem III. will be the longitude of the place of observation in time, supposing the lunar and solar tables to be correct; but it is much more accurate to compare actual observations made at different places, by deducing the times of the ecliptic conjunction from each observation, the difference of these times will be the difference of longitude. EXAMPLE. At Salem, in the latitude of 42° 33′ 30′′ N. longitude by estimation 4h, 43′ 32 W. from Greenwich, the beginning of the total eclipse of June, 1806, was observed at 15d. 22h, 6 18.1 apparent time by astronomical computation Required the longitude of the place from this observation? The elements must be calculated as in the example of Problem VI. for the beginning of the eclipse, except those marked in italics. The rest of the calculation may be made by proportional logarithms as follows: The longitude and latitude must be corrected for the errors of the tables, when known, by a previous operation, or by other observations. †This correction must be found after the altitude and longitude of the nonagesimal are calculated. These calculations may be made in the same manner by using proportional logarithms, the only dif ference consists in using the constant logarithm 04771 instead of 3.55630 in finding the time of conjunction. In general, the beginning of an eclipse or occultation precedes the apparent conjunction, and the end is after the apparent conjunction, but there is a case (which very rarely occurs) where the contrary may take place: namely, where th· point For L. (P. XII. fig. 12, 15) falls between C and B, which can happen only when the lines FD, EL are nearly equal to SF or SL. In this case it may be ascer tained whether the phase precedes or follows the conjunction by making the calculation as in Prob. VI. or VII. with the times of beginning and end, calculated by Problem XIII. and as the central angle is greater or less than 90%, the phase will follow or precede the apparent conjunction. The latitudes given by the tables being supposed current. on it as before. Whence the times of beginning and end of the eclipse m as in the above rule. An example of this method is not given, as it would scheme too confused. PROBLEM XII. To project an occultation of a fixed star by the moon, at any given pla The method of projecting an occultation is nearly the same as that of a the sun, but to save the trouble of reference it was thought expedient to without abridgment. RULE. To the time of the ecliptic conjunction of the moon and star, given in th of the Nautical Almanac (or calculated by Prob. III.) add the longitude of t place turned into time, if east, but subtract if west, the sum or difference will of conjunction at the proposed place. Corresponding to the time of co Greenwich, find by Problem 1. the moon's latitude, horizontal parallax and se also the sun's right ascension., Then by Problem II. find the horary motion in longitude and latitude, and by Tables VIII. and XXXVII. the star's Right Declination, Longitude and Latitude. T Draw the line ACB (Plate XII. fig. S.) representing a parallel of the ecli through the star, and perpendicular thereto the line CPR. Take a scale of to measure the lines of the projection, and from it take an interval equal to ence of the latitudes of the moon and star, and apply it to the line CR fi above the line ACB if the moon's latitude is north of the star's, otherwi Take CO equal to the horary motion of the moon in longitude, and set it on to the right hand of C to O; take CP equal to the moon's horary motion found with its sign by Problem II. and set it on the line CR from C to P, line ACB, if its sign is, below if +. Join OP which represents the ho of the moon on her orbit, and parallel to that line draw the orbit of the moor which are to be marked the places of the moon before and after the con means of the horary motion OP, so that the moment of the ecliptic at the proposed place may fall exactly at the point G, as in the figure conjuration is at 18h. 42'. This may be done by making OP equal to the distance 60, 60, on the line of lines of the sector, then measuring from the sar transverse distance corresponding to the minutes and parts of a minute of the ecliptic conjunction at the place of observation, and setting it GN from G towards the right to the point x, the place of the moon at the hourt preceding the conjunction (which in the present figure is 18h.) Then OP being taken in the compasses, and set from r to the right hand, gives succ preceding hours, and the same distance set to the left gives the following the figure, where they are marked 17h. 18h. 19h. 20h. These hours are to into 60 equal parts representing minutes, the scale being taken sufficiently la purpose. In the present figure the subdivisions are carried only to five min the moon's horizontal parallax from the scale of equal parts for the radius which on the centre C, describe the circle BRA cutting CR in R. Open th the transverse distance 60°, 60°, on the line of chords, is equal to the radi measure from that line the transverse distance 23° 28′ (equal to the oblig ecliptic) which set on the circle ARB on each side of R to T and U. Join CR in Q. On Q as a centre, with the radius QT, describe a circle TYUV, 0 off the arch TYV, equal to the star's longitude. Through V draw the line to CR. Open the sector till the transverse distance 90°, 90°, on the sines, is radius CB, then take in the compasses from the same lines an extent equal t verse distance corresponding to the complement of the declination of the st one foot in C sweep a small arch to cut the line VP' in P' the place of the earth.*f Draw CP', and continue it on either side so as to cut the in the point W situated above AB, the latitude of the proposed place is below if south. In the proposed figure the latitude is north. (If it had the lower part of the circle ARB ought to have been made use of.) Open In strictness these quantities ought to be corrected for Aberration and Nutation by Tat -XLIII. but the correction is so small that it may always be neglected. If the Right A Declination only are given, the latitude and longitude may be found by Problem XIX fatter are given, the former may be calculated by Problem XX. **In the figure the point G is placed above ACB, because the moon is in a less sout than the star. This part of the rule may also be thus expressed. Find the moon's lati sign as in Problem II. Prefix the sign to the star's latitude if north, the sign-i the latitudes, noticing the signs as in algebra, and the distance CG will be obtained. Il the point G is to be placed above C, but below C if the sign is See note with this mark in page 592. See note with this mark in page 592. The distance of the fine WV from the line CR, the situation of the point I and the spectator, may be found as in the note § page 594. 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 1, q, n. Bisect In in r, and erect the line t r u 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 tor t, and on each side of r mark on the line tru the sines of 15°, 30°, 45°, 60°, 75°, (equal to 1h. 2h. Sh. 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 I 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 ntlu'n 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 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 I of the path is taken for the hour of passing the meridian 19h. 54. The extremities of the dotted lines to the left being marked successively 20h. 54′, 21h. 54', 22h. 54′, 23h. 54', Oh. 54', &c. The path touches the circle AR in two points, representing the points of rising and setting of the star, which in the present figure are 14h. 9 and 1h. 39. 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. 24', which is 3h. 30' or 52 30′ from the time 7h. 54', 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 37 (which is the complement of 524) 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 semidiameter, describe the small circles mecting 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 lines 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 Fig. 9, Plate XII. in which the point C, p' Z', are similarly situated to the corresponding points of Fig. 8, marked with the same letters. Through p' draw the line a' pc' parallel to CZ', to meet the moon's disc in a', c'. Then the circled 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 d' 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, e' 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 timate 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 p. 0, of the moon may also be made use of for this purpose. The line pp) 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 • Or rather the Lorary distance of the and at the time of the ecliptic conjunction of the moon and star. |