sum, rejecting 10 in the index, will be the log. of the difference of the segments FB, BL; half of which added to, and subtracted from half of FL, will give the two segments FB, BL, the greater seginent being contiguous to the greater side, whether SF or SL. Then from the logarithm of the segment FB, increasing its index by 10, subtract the logarithm of SF, the remainder will be the log. sine of the angle FSB, which is always less than 90°. The difference between this and the angle of inclination DSB, will be the central angle DSF. To the log. co-sine of the central angle, add the logarithm of the moon's corrected semi-diameter at the immersion SF, and the log. secant of the star's latitude, the sum, rejecting 20 in the index, will be the logarithm of the apparent difference of longitude of the moon and star at that time. This is to be subtracted from the true longitude of the star, if the central angle is less than 90°, but added, if greater than 90, the sum or difference will be the moon's apparent longitude; to this must be added the moon's parallax in longitude, when her distance from the Nonagesimal (found as in Problem V. by subtracting the longitude of the Nonagesimal from the moon's longitude, borrowing 360 when necessary) is greater than 150, otherwise the parallax must be subtracted; the sum or difference will be the moon's true longitude at the beginning of the occultation. Take the difference in seconds between the true longitudes of the star and moon at the beginning of the occultation, to the logarithm of which, add the arithmetical comp. log. of the moon's horary motion in seconds, and the constant logarithm 3.55630, the sum, rejecting 10 in the index, will be the logarithm of the time from the conjunction in seconds, which is to be added to the observed apparent time of the beginning of the occultation, when the star's longitude is greater than the moon's true longitude at that time, otherwise subtracted; the sum, or difference, will be the apparent time of the true ecliptic conjunction of the star and moon at the place of observation. The difference between this and the time of conjunction, inferred from the Nautical Almanac by Problem III. for the meridian of Greenwich, will be the longitude of the place. If corresponding observations be made at different places, it will be much more accurate to deduce from them the time of conjunction at each place, and take the difference of those times for the difference of meridians, if it does not differ much from the supposed difference of longitude. If there is considerable difference, the operation must be repeated, making use of the longitude found by this operation; and thus by successive opera-> tions the true longitude may be obtained. The long. of the place of observation being accurately known, the errors of the lunar tables in lat. and long. may be easily found. For the difference between the moon's true longitude, deduced from the observations by the above method, and the longitude found from the Nautical Almanac, will be the error of the tables in longitude. To find the error in latitude proceed thus: To the log. sine of the central angle DSF, add the logarithm of the corrected semi-diameter of the moon at the immersion SF, the sum, rejecting 10 in the index, will be the logarithm of the apparent difference of latitude of the moon and star, which added to the true latitude of the star, with the sign + if the point F falls below the line DE, but with the sign — if above, will give the apparent latitude of the moon at that time, the difference between this and the apparent latitude found by Problem V. will be the error of the tables, always supposing the sign + to be prefixed to southern latitudes, the sign — to northern, and noting the signs as in algebra.* REMARK. In the two preceding Problems the time of the true conjunction is calculated by means of the triangle SFD, but it will be useful for the purpose of verification, to go over the calculation by means of the triangle SLE. The process is nearly the same in both methods. The differences consist in finding the angle LSB by subtracting the logarithm of SL from the logarithm of LB, increasing its index by 10, the remainder will be the log. sine of the acute angle LSB, which, added to the angle of inclination, (found as before) will give the central angle DSL, with which and the distance SL, corresponding to the end of the eclipse or occultation, may be found the apparent diff. of longitude between the sun and moon, and moon and star; this is to be added to the longitude of the sun or star at that time, if the central angle exceed 90, otherwise subtracted, the sum or difference will be the apparent longitude of the moon corresponding, from which the time of the ecliptic conjunction may be obtained as before. If the central angle exceed 180° the sine and co-sine of the excess of that angle above 180 must be found instead of the sine and co-sine of the central angle. The apparent latitude of the moon is found as in the preceding rules, by making use of the central angle DSL and the valute SL, corresponding to the end of the ecl pse or occultation; whence may be deduced the apparent latitude and the error of the tables in latitude. It is evident that both these methods ought to give the same results and thus furnish a proof of the correctness of the calculations. All these calculations may be made by proportional logarithms, by reading in the rule, log, co-tang for log, tang, log, co-secant for og sine, &c. as was mentioned at the end of the rule in Problem V. and by using the constant lo; 0.1771 instead of 3 "5630. When SF=SL the angie ay be found as in the note, with this mark in page 583. When this varies, it must be taken to correspond to the middle time between the immersion and true conjunction. * See note with this mark in page 584. EXAMPLE. Suppose in a place in the latitude of 20° 0' N. longitude 1h. 9m. Os. east of Greenwich by estimate the occultation of Spica by the Moon on Dec. 12, 1808, was observed; the immersion at 16h. 57 29', eer. sion at 18h. 10' 29", apparent time by astronomical computation. Required the longitude of the place of observation? Most of the elements in the following Table are calculated by Problems I. II. and VI. The difference of the apparent latitudes of the Moon and Star at the beginning of the occultation 3' 21'.3 being less than at the end 6' 54.2 the angle of inclination is less than 90°. In this example the moon's latitude is more southerly than the star's, hence the points F, L, fall below the line DE. Diff. app. long. D 31' 8' 8=1868.8 D's mean app. lat. 2 7 21 log. 3.27156 cos. 9.99970 The moon's horary motion varies from 35' 51'.7 to 35′ 54′′.2 during the occultation, hence at the middle time 17h. 49′ 45′′ between the immersion 16h. 57' 29" and the conjunction 18h. 42′ (deduce from the Nautical Almanac) the horary motion was 35' 53.5 as is easily found by a calculation similar to that in the Example of Problem VI. The difference of meridians deduced from the observation 1b. 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. 90". 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 time 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 sun by Problem II. Decrease the sun's semi-diameter 5 for irradiation. Decrease the moon's semi-diameter 2' for inflerion, 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 logaricans 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 dirence 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 180, 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 3.55630, 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. 45' 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 3.55030 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 the 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 ascertained 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 30°, the phase will follow or precede the apparent conjunction. The latitudes given by the tables being supposed correct. Diff. Merid. 1h. 31' 15" 15 22 6 18 15 23 87 31 4.41 29 by N. A. f we suppose the time of conjunction at Greenwich to be 4h. 20′ 47', as calculated in the expe Problem VI. the difference of meridians would be 4h. 43' 16'', agreeing nearly with the assumed fungi. tude, so that it will not be necessary to repeat the operation. The remarks at the end of that examp respecting the errors of the lunar tables, and the comparing of actual observations at different places, are equally applicable to the present Problem. PROBLEM IX. To find the longitude of a place by an occultation of a fixed star by the moon, when the immersion or emersion only is observed, the apparent time being estimated from noon to noon, according to the method of astronomers, and the latitude of the place being knowu. RULE. * To the apparent time apply the estimated longitude of the place turned into time, by adding if west, subtracting if east, the sum or difference will be the supposed time at Greenwich. At this time find in the Nautical Almanac the sun's right ascension, the moon's semi-diameter, horizontal parallax, longitude and latitude by Problem 1. and the moon's horary motion by Problem II. also the latitude and longitude of the fixed star by Table XXXVII. and correct it for aberration and equation of equinoxes by Tables XL. XLI. Decrease the moon's semi-diameter 2" for inflexion, and to the remainder add the augmentation from Table XLIV. † the sum will be the corrected semi-diameter. Find also, in the Nautical Almanac, the obliquity of the ecliptic. With these ments and the apparent time of observation, calculate the altitude and longitude of the Nonagesimal by Problem IV. also the parallaxes in longitude and latitude of the moon's apparent latitude by Problem V. Take the difference between the latitude of the star and the app. lat. of the mock which add to, and subtract from the moon's corrected semi-diameter (these quant ties being expressed in seconds) half the sum of the logarithms of these quantities i creased by the log. secant of the star's latitude, rejecting 10 in the index, will be the logarithm of an arch in seconds to be added to the star's longitude if the moon bas passed the apparent conjunction, but subtracted if before, § the sum or difference will be the apparent longitude of the moon. when the moon's distance from the nonagesimal (found as in Problem VII. by To this add the moon's parallax in longitude 'tracting the longitude of the nonagesimal from the moon's longitude, borrowing 360when necessary) is greater than 180°, otherwise subtract it, the sum or difference will be the true longitude of the moon. Take the difference in seconds between the moon and star's true longitudes, and to its logarithm add the arithmetical comp. log. of the moon's horary motion and the constant logarithm 3.55630, the sum, rejecting 10 in the index, will be the logarithm of a correction in seconds to be applied to the than the star's, otherwise subtracting, the sum or difference will be the time of the true cof junction at the place of observation. The difference between this and the time of conjunction inferred from the Nautical Almanac by Problem III. for the meridian of Greenwich, will be the longitude of the place of observation, if the tables are co rect; but it is much more accurate to compare the times of conjunction deduced * Corrected for the errors of the tables in longitude or latitude when known. This correction must be found after the altitude and longitude of the nonagesimal are calculated. § See note with this mark in page 589. 0,4771 instead of 5,55630, and the log. co-sine being used instead of log. sceant. Proportional logarithms may be used instead of common logarithms, the constant logarithm leng from actual observations at the different places in the manner mentioned at the end of the rule given in Problem VII. EXAMPLE. Suppose in a place in the latitude of 20° 0' N. longitude by estimation 1h. 9′ 0′′ east from Greenwich, the emersion of the star Spica was observed on December 12, 1808, at 18h. 10 29, apparent time by astronomical computation. Required the longitude of the place of observation? The elements must be calculated as in the example of Problem VII. for the emersion of Spica. The rest of the calculation, made by common logarithms, is as follows. D Semi-diameter 16' 30 .8930 ́.8 Diff. * longitude app. long. par. long. D true long. Time 15' C 6 201 10 30 .7 201 25 SI S 200 51 S7 3 Time of obs. 18 10 29 13 53.4 1133.4 31 31 =1834 The difference of meridians by calculation 1h. 9' 3' differs but 3" from the assumed longitude, so that it will not be necessary to repeat the operation. All the remarks made at the end of the example in Problem VII. are applicable to this problem. It may also be further observed, that the emersion or immersion which happens on the dark limb of the moon can be observed with much more accuracy than on the enlightened limb; because the light from this limb prevents the observer from perceiving the star's immersion or emersion so instantaneously as on the dark side of the moon. PROBLEM X. To calculate an eclipse of the moon. The time of beginning or end of a lunar eclipse at any place may be found by subtracting or adding the longitude to the times given in the Nautical Almanac for the ineridian of Greenwich, according as the longitude is west or east. But as some readers may wish to know the method of deducing these times from the longitudes, latitudes, &c. of the moon and sun, given by the Nautical Almanac or by other tables, it was thought proper to insert the rule for these calculations. An eclipse of the moon can only happen at the time of the full moon. If her longitude at that time is not distant from either node* of the moon's orbit more than about 12, there may be an eclipse. To find whether there will be one, and to calculate the times and phases, proceed as follows. RULE. Find the time of full moon at Greenwich by the Nautical Almanac or Problem III. to which add the longitude of the place turned into time if east, but subtract if west, the sum or difference will be the time of the ecliptic opposition at the proposed place. For the time at Greenwich find by Problem 1. the moon's latitude, horizontal parallax and semi-diameter (which requires no augmentation) also the sun's semi-diameter. Then by Problem II. the horary notion of the moon, from the sun in longitude, and the moon's horary motion in latitude. Draw the line ACB (Plate XII. fig. 6) and perpendicularly thereto the line PCR. Select a scale of equal parts to measure the lines of projection, and from it take CG equal to the moon's latitude, and set it on CR from C to G, above the line AB if the latitude of the moon is north, below if south. Take CO equal to the horary motion of the moon from the sun in longitude, and set it on the line CB to the right of *The longitude of the mon's asending node is given in the thi & page of the Nautical Almanac. The longitude of the other node is found by adding subtracung 6 sites. + The northern tatitudes found by Prob 1. have the sign, the southern +. In the figure the latitude is south. If it had been north the point G must have been placed on the continuation of RC above C. |