the rule, beginning and end of the internal contacts, instead of beginning and end of the eclipse, and taking SF, SL, equal to the differences of the corresponding semi-diameters, instead of their sums.

EXAMPLE.

At Salem, in the latitude of 42° 33′ 30′′ N, longitude by estimation 4h. 43m. 32s. W. from Greenwich, the beginning of the total eclipse of June, 1806, was observed at 15d. 22h. 6m. 188.1, and the end at the 16d. Oh. 50m. 34s.6, apparent time, by astronomical computation. Required the longitude of the place of observation.

Most of the following elements are calculated in Problems I. II. IV. V.

As the apparent latitude at the beginning of the eclipse is north, and at the end south, the point F corresponding to this example falls above DE, the point L below it. The rest of the calculation is as follows:

The mean parallax formerly used was 8.8: it is now found to be nearly 8".6. †This horary motion increases from 34 16.1 to 34′ 19′′.7, or 3.6, during the eclipse 2h. 44m. 168.5, which Is 1.32 per hour. Now the ecliptic conjunction, or time of new moon, at Greenwich, by the N. A., was 4h. 19m, or rather 4h. 20m. 47s., corresponding to 23h. 37m. 15s. at Salem, which is 1h. 30m. 578. after the beginning of the eclipse; and the increase of the horary motion in half that time is 1", which, added to 34 16.1, gives the horary motion 34/ 17".1, corresponding in the middle time between the beginning of the eclipse and the conjunction. This is used in calculating the correct time of conjunction. We may remark that, in the above calculations, we have used the apparent times of observation, to conform to the arrangement of the Nautical Almanac in 1806; but in the present form of the Nautical Almanac, it will be convenient te

use the mean time.

In finding the time of conjunction or new moon, at Greenwich, 4h. 19m., in the Nautical Almanac, the longitude of the moon was supposed to be given correctly by the tables. If the calculation be made by Problem III., after allowing for the error -585, the result will be 4h. 20m. 47s., whence the difference of meridians = 4h. 43m. 10s.8, which differs so little from the assumed longitude, 4h. 43m. 32s., that it will not be necessary to repeat the operation. If the eclipse was observed at Greenwich, the time of conjunction ought to be determined thereby, in a similar manner to the above calculations; or by those of Problem VIII., if only one of the phases is observed: by this means the errors of the tables will be wholly avoided. If the eclipse was not observed at Greenwich, the observations at any other place whose longitude is known might be made use of, and thus the difference of meridians accurately obtained. The moon's true longitude, deduced from the above observation, is 83° 48′ 5371; by the Nautical Almanac it is 83° 50′ 2.0; the difference, - 68.9, would be the error of the tables by this observation, if the assumed longitude, 4h. 43′ 32, and the solar tables, were correct. By repeating the operation with the assumed longitude, 4h. 43m. 10s.8, the error, 68.9, would be reduced to nearly the estimated value, 58.5.

The eclipse was so nearly central at Salem, that a variation of a minute in the moon's latitude would hardly alter the times or duration of the eclipse; so that the latitudé could not be determined by the above observations to any considerable degree of accuracy. From this cause it happens that the apparent latitude at the beginning of the eclipse is by the above calculation 2.8, instead of 1/ 55.8, as found by allowing the error, 11.4, deduced from other observations made where the eclipse was not so nearly central, and by the limits of the shadow of total darkness.

PROBLEM VII.

Given the latitude of the place, and the apparent times of the beginning and end of an occultation of a fixed star by the moon, to find the longitude of the place of observation. In the following rule, reference will be made to figure 13, Plate XIII.,in which DSE repre sents a parallel to the ecliptic passing through the place of the star S; SF, SL, the corrected semi-diameters of the moon at the beginning and end of the occultation; DF, EL, the dif ferences between the apparent latitudes of the moon and the star, when of the same name, or their sums, when of different names; either of these lines falling below DE if the moon's apparent latitude is more southerly than that of the star, otherwise above.

RULE.

To the apparent times of the beginning and end of the occultation, add the estimated longi. tude of the place in time if it is west, but subtract if east: the sum or difference will be the supposed time at Greenwich; corresponding to which, in the Nautical Almanac, find, by Problem I., the moon's semi-diameter, horizontal parallax, longitude and latitude,* and the sun's right ascension; also the moon's horary motion by Problem II., and the true longitude and latitude of the fixed star, by Table XXXVII., corrected for aberration and equation of equinoxes by Tables XL., XLI. This may also be deduced from the right ascension and declination of the star, if it be given in the Nautical Almanac, by means of Problem XIX. of this Appendix. Find, also, in the Nautical Almanac, the obliquity of the ecliptic. To the moon's semi-diameter, add the correction in Table XLIV.,† and from the sum subtract the inflexion, 2", if it be thought necessary; the remainder will be her corrected semi-diameter. With these elements and the apparent times of the place of observation, calculate the altitudes and longitudes of the nonagesimal, by Problem IV., and the parallaxes in longitude and latitude, and the moon's apparent longitudes and latitudes, by Problem V.

Take the difference between the apparent longitudes of the moon at the beginning and end of the occultation, which will be the moon's apparent motion in longitude, the logarithm of which, in seconds, being added to the log. cosine of the mean of the apparent latitudes of the moon at the beginning and end of the occultation, rejecting 10 in the index, will be the logarithm of the motion of the moon on the parallel FA. The relative motion in latitude AL is found by taking the difference of the moon's apparent latitudes at the beginning and end of the eclipse if they are both north or both south; but their sum if one be north and the other south. From the logarithm of FA, increasing the index by 10, subtract the logarithm of AL; the remainder will be the log. tangent of the angle of inclination DSB; this angle is to be taken greater than 90° when the difference of the moon's and star's apparent latitudes at the beginning of the occultation FD is greater than at the end EL, otherwise less. § Then to the log. cosecant of the angle of inclination, add the logarithm of the relative motion FA; the sum, rejecting 10 in the index, will be the logarithm of the apparent motion of the moon in her orbit FL.

*Corrected for the errors of the tables in longitude and latitude, when known.

†This correction must be found after the altitude and longitude of the nonagesimal are calculated.

The mean latitude is half the sum of the two latitudes, if they are of the same name, but their half differ ence, if of different names. In solar eclipses, the correction for the mean latitude of the moon is neglected as too small to be taken notice of, the distance FA being taken equal to the difference of longitude DE (fig. 12. Plate XIII.).

This rule is equally true, whether the points F, L, fall on the same or on different sides of the line DE. If DF, EL, are equal, and the points F, L, fall on the same side of DE, the angle DSB will be 90°. If they are qual, and those points fall on different sides of the line DE, the angle DSB may be taken acute or obtuse. in strictness, when the points F, L, fall on different sides of DE, the angle DSB is greater or 'ess than 90°,

FD

according as the quantity is greater or less than

EL

Then, in the triangle SFL, the sides SF, FL (representing the corrected semi-diameters of the moon at the immersion and emersion), and the relative motion FL, are given to find the angle FSB (by Case VI. Oblique Trig.). Thus: to the log. arith. comp. of FL, add the logarithm of the sum of SF and SL, and the logarithm of their difference: the sum, rejecting 10 in the index, will be the logarithm of the difference of the segments FB, BL; half of this, being added to, or subtracted from the half of FL, will give the two segments FB, BL; the greater segment 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 dif ference between this and the angle of inclination DSB, will be the central angle DSF.

To the log. cosine of the central angle add the logarithm of the moon's corrected semidiameter 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 180°, 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 this 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 operations, the true longitude may be obtained.

The longitude of the place of observation being accurately known, the errors of the lunar tables in latitude and longitude 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, being 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, being added to the angle of inclination (found as before), will give the central angle DSL: with this, and the distance SL, corresponding to the end of the eclipse or occultation, may be found the apparent difference 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 cosine of the excess of that angle above 180° must be found instead of the sine and cosine 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 value SL, corresponding to the end of the eclipse 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. cotangent for log. tangent, log. cosecant for log. sine, &c., as was mentioned at the end of the rule in Problem V., and by using the constant log. 0.4771, instead of 3.55630.

*When SF = SL, the angle may be found as in the note with this mark in page 408.

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 408.

EXAMPLE.

Suppose in a place in the latitude of 20° 0' N., longitude 1h. 9m. Os. east of Greenwich, by estimation, the occultation of Spica by the moon on December 12, 1808, was observed; the immersion at 16h. 57m. 29s., emersion at 18h. 10m. 29s., 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.

Difference apparent long. D 31' 8".8=1868".8
D's mean apparent lat.

Distance FA

Log. 3.27156

2 7 21

Cosine 9.99970

Log. 13.27126

D's difference lat..... AL-3 32.9 = 212.9

Log. 2.32818

Log. 3.27196 Cosecant 10.00280

Log. 3.27406

Log. 2.99445
Sec. 10.00027

FD 199.9 =

Log. 2.98563

#'s latitude

201 10 30.7
200 54 23.2 by observation.
46 25
200 7 58.2

Difference true longitude 3752.5=1 2 32.5

Constant 3.55630
Log. 3.57432

D's horary motion....... 2153.5= 35 53.5 Ar.Co. Log. 6.66686

62731h. 44m. 33s.

............................Sine 9.30643 .....Log. 2.99445

3 19.9 Log. 2.30083 22 13.3

2 5 33.2 by obs.
2 5 34.6 by N. A.

D's app. lat.
D's app. lat.
Error Table
1.4 in latitude.
D's true lon. 200 7 58 2 by obs.
D's true lon. 200 7 56 .3 by N. A.
Error Table + 1.9 in longitude.

Log. 3.79748

16 57 29

18 42 2 at place of observation.
17 33 0 at Greenwich.

1h. 9m. 2s.

The moon's horary motion varies from 35' 51.7, to 35' 54".2, during the occultation: hence, at the middle time, 17h. 49m. 45s., between the immersion, 16h. 57m. 29s., and the conjunction, 18h. 42m. (deduced from the Nautical Almanac), the horary motion was 35' 53.5, as is easily found by a calculation similar to that in the

The difference of meridians deduced from the observation, 1h. 9m. 2s., differs but 28. from the assumed quantity, 1h. 9m. Os. 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. 9m. Os. 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 longitude of the moon, deduced from the calculation, will agree also with the longitude by the tables. The time of conjunction at Greenwich, 17h. 33m. Os., 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, subtracting 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 34" for irradiation. Decrease the moon's semi-diameter 2" for inflexion, if it be thought necessary, 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 logarithms will be the logarithm of an arc 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 difference, 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. 43m. 32s. W. from Greenwich, the beginning of the total eclipse of June, 1806, was observed at 15d. 22h. 6m. 18s.1,

* 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 difference consists in using the constant logarithm 0.4771, instead of 3.55630, in finding the time of conjunction. In general, the beginning of an eclipse or occultation precedes the apparent conjunctior, 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 (Plate XIII. fig. 12, 13) 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 Problem 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

correct.