EXAMPLE I.

Required the altitudes and longitudes of the nonagesimal at Salem, June 16, 1806, at the times of the beginning and end of the eclipse, calculated in Problem VI.

Required the altitudes and longitudes of the nonagesimal at the times and places mentioned in the Example of Problem VII.

In these calculations, it is usual to take the sun's right ascension, and the apparent times, to tenths of a second, and to take proportional parts for the seconds and tenths in finding the logarithms. Thus, in Example I., in finding the log. cotangent of 9h. 43m. Ss.1, the nearest logarithms are 9.48849, 9.48804, corresponding to the times 9h. 43m. 4s., 9h. 43m. 12s. These logarithms differ 45, the times 8s.; and the difference between 9h. 43m. 4s., and 9h. 43m. 8s.1, is 4s.1. Hence, 8s.; 45: 48.1: 23, the correction to be subtracted from the first log. 9.48849 (because it is decreasing), to obtain the sought log. cotangent 9.48826.

PROBLEM V.

Given the altitude and longitude of the nonagesimal; the longitude, latitude, and hori zontal parallax of the moon, and the latitude of the place of observation; to find the moon's parallax in latitude and longitude.

RULE BY COMMON LOGARITHMS.

From the horizontal parallax of the moon, subtract its correction from Table XXXVIII, corresponding to the latitude of the place; the remainder, in occultations of a fixed star, will be the reduced parallax; but in solar eclipses, this quantity is to be diminished by the sun's horizontal parallax, 8".6,* to obtain the reduced parallax.

To the logarithm of the reduced parallax in seconds, add the log. sine of the altitude of the nonagesimal, and the log. secant of the moon's true latitude; the sum, rejecting 20 in the index, will be a constant log. From the moon's true longitude, increased by 360° if necessary, subtract the longitude of the nonagesimal; the remainder will be the moon's distance from the nonagesimal, which, if less than 180°, is to be called the arc D, otherwise its supplement to 360° is to be called the arc D. To the constant logarithm add the log. sine of D; the sum, rejecting 10 in the index, will be the logarithm of the approximate parallax in longitude in seconds, which add to the arc D; then take the log. sine of the sum, and add it to the constant logarithm, rejecting 10 in the index, and the logarithm of the corrected parallax will be obtained. This will, in general, be sufficiently exact; but when great accuracy is required, the operation may be again repeated, by adding the arc D to the corrected parallax ; then to the log. sine of the sum add the constant logarithm, rejecting 10 in the index, and the logarithm of the parallax in longitude P will be obtained. This is

parallaxes. It is immaterial whether the altitude of the nonagesimal, or its supplement, is made use of in Table XLIV.

*This is nearly the mean value of the sun's parallax; but it will be more accurate to use the actual value an it is given in page 266 of the Nautical Almanac.

† Corrected for the errors of the tables, when known.

This sum Dcor. par. is nearly equal to D+P, the apparent distance of the moon from the nonagesi

to be added to the true longitude of the moon when her distance from the nonagesimal is less than 180°, otherwise subtracted to obtain her apparent longitude.

If the true latitude of the moon is south, prefix the sign + to it; if north, the sign. Then to the logarithm of the reduced parallax in seconds, add the log. cosine of the altitude of the nonagesimal, and the log. cosine of the moon's apparent latitude;* the sum, rejecting 20 in the index, will be the logarithm of the first part of the parallax in latitude in seconds, to which prefix the sign when the altitude of the nonagesimal is less than 90°, otherwise the sign; this being added to the true latitude of the moon, due regard being paid to the signs, will give her approximate latitude.

To the logarithm of the reduced parallax in seconds, add the log. sine of the altitude of the nonagesimal, the log. sine of the moon's approximate latitude, and the log. cosine of the sum of the arcs D and P; the sum, rejecting 30 in the index, will be the logarithm of the second part of the parallax in latitude in seconds, to which prefix the sign - when the arcs DP, and the approximate polar distance,f are both greater or both less than 90°, otherwise the sign+; this term, being connected with the approximate latitude, will give the apparent latitude of the moon, which will be south if +, north if — The moon's true latitude subtracted from her apparent latitude, noticing the signs, will give the parallax in latitude.

BY PROPORTIONAL LOGARITHMS.

The above rule will answer in calculating by proportional logarithms, with the following alterations. When the log. sine occurs, read log. cosecant; for log. cosine, read log. secant; for log. secant, read log. cosine; and for log. cosecant, read log. sine. The parallaxes may be calculated to the nearest second by proportional logarithms. When greater accuracy is required, common logarithms must be made use of.

To illustrate this rule, the following examples, corresponding to the times of the beginning and end of the total eclipse of the sun, of June 16, 1806, as observed at Salem, are given. The elements necessary for this purpose have already been calculated in Problems I. and IV. For greater accuracy, the longitudes and latitudes of the moon are corrected for the errors -58.5 in longitude, and 11".4 in latitude, which were found by comparing several observations of the eclipse made at different places.

-

EXAMPLE I

Given the altitude of the nonagesimal 67° 58′ 50′′, its longitude 63° 22′ 31"; the longitude of the moon 83° 49′ 3.5, her latitude 24' 27" 4 N., her horizontal parallax 60' 24".1; the latitude of the place of observation 42° 33′ 30′′; required the parallaxes in longitude and latitude.

The correction in Table XXXVIII. corresponding to the latitude 42° 33′ 30′′, and parallax 60 24.1, is 5.6; this, and the sun's horizontal parallax, 8.8, subtracted from the moon's horizontal parallax, 60' 24.1, leaves the reduced parallax 60′ 9′′.7=3609".7. The longitude of the nonagesimal, 63° 22′ 31", subtracted from the moon's longitude, 83° 49' 3", leaves the moon's distance from the nonagesimal, 20° 26′ 32′′, equal to the arc D, because it is less than 180°.

* In solar eclipses, the apparent latitude is so small that its log. cos. may be put equal to 10.00000. In occultations, you must calculate the first part of the parallax in altitude hy approximation, making use of the true latitude instead of the apparent in the above rule, and deducing the approximate value of the first part of the parallax; this applied to the true latitude will give the approximate apparent latitude, with which the operation is to be repeated, and the first part of the parallax will be obtained to a sufficient degree of

exactness.

The apparent polar distance is found by adding +90° to the approximate latitude, due regard being had to the signs. To be perfectly accurate, the apparent instead of the approximate latitude ought to be made use of in this part of the calculation, and the logarithms of this term ought to be increased by the log. secant less radius of P; but these corrections are too small to affect the result. In calculating the second part of the parallax in latitude, it will be sufficient to take the logarithm to three or four places of the decimals.

This rule gives the apparent latitude in all cases; but it may not be amiss to observe, that, in several late publications, the cases where the moon is between the zenith and the elevated pole are by mistake neglected.

EXAMPLE II.

Given the altitude of the nonagesimal 70° 57′ 46", its longitude 95° 26' 36"; the longitude of the moon 85° 29′ 32.6, her latitude 15' 10".4 N., her horizontal parallax 60′ 27′′.0; the latitude of the place of observation 42° 33′ 30′′; required the parallaxes in longitude and latitude.

The correction in Table XXXVIII., corresponding to the latitude 42° 33′ 30", and paral lax 60' 27", is 5.6; this, and the sun's horizontal parallax, 8.8, subtracted from the moon's horizontal parallax, 60' 27".0, leaves the reduced parallax 60' 12.6. The longitude of the nonagesimal, 95° 26' 36", subtracted from the moon's longitude increased by 360°, viz. 445° 29′ 33′′, leaves the moon's distance from the nonagesimal 350° 2′ 57′′, the supplement of which to 360° is 9° 57′ 3′′, equal to the arch D.

1 55 11

EXAMPLE III.

Required the parallaxes in longitude and latitude at the time of the occultation of Spics, December 12, 1808, at the times and places mentioned in the Example of Problem VII.

Reduced parallax
Alt. nonagesimal
D's true latitude

Cosecant 10.0050
Cosine 9.9998

..Secant
Secant

0.4782

D's app. latitude*
1 part par. lat.
D's true lat.

10.8199 10.0003

+

+1 55 11.0

Alt. nonagesimal
D+P

51 15 13

D's true longitude 200 7 56.3

Prop. Log. 0.4789
Cosecant 10.0050
Secant 10.2035

Prop. Log. 2.1988

+25 34.6 South.

10 23,6

*The moon's true latitude, 1° 55' 11", must first be used, its log, secant being 10.0002, which give the let part parallax 9 3, which, added to the true latitude of the moon, gives the approximate latitude nearly 24/14", the log. secant of which is 10.0003, as above. The calculation for the emersion is made in a similar

D's approx. latitude
1 part par. lat.
+ 15' 54".2
D's true lat. +1 51 29.1

D's approx. lat. +2 7 23.3

Reduced parallax
4937
Alt. nonagesimal
D+P

Cosecant 11.4313

35 39 35

Prop. Log. 0.4780
Cosecant 10.0159
Secant 10.0902

2 part par. lat.

+ 1 44.2
D's approx. lat. +2 7 23.3
D's appar. lat.
9
+ 2 7.5 South.
D's parallax lat. + 17 38.4

Prop. Log. 2.0154

Having thus explained the method of calculating the parallaxes of the moon, it now remains to give the rules for finding the longitude by eclipses and occultations. The main object in these calculations is to determine, from the observed beginning or end of the eclipse or occultation, the precise time of the ecliptic conjunction of the sun, or star and moon, free from the effects of parallax, counted on the meridian of the place of observation, since the difference of the times of conjunction, obtained in this manner at two places, will be their difference of longitude. If the lunar and solar tables were perfectly correct, the longitude might be determined by taking the difference between the time of conjunction given in the Nautical Almanac, and that deduced from the observations of the eclipse or occultation; but it is much more accurate to compare the times deduced from observations actually made at the places for which the difference of longitude is sought. There are two different methods of finding the ecliptic conjunction, according as the latitude of the moon is supposed to be accurately known or not. If the latitude was given correctly by the lunar tables, or was accurately known by other observations, the ecliptic conjunction, and the longitude of the place, might be determined by each of the phases of the eclipse or occultation, by the method given in Problems VIII. and IX. But the moon's latitude not being generally given to a sufficient degree of accuracy, it is usual to combine together the observations of the beginning and end of the eclipse or occultation, or the beginning and end of total darkness in a total eclipse, or the two internal contacts of an annular eclipse, to ascertain the error of the moon's latitude, by the method given in Problems VI. and VII. In making the calculations in these Problems, it will be necessary to know nearly the longitude of the place, in order to find the supposed time at Greenwich, so as to take out the elements from the Nautical Almanac; and if the longitude deduced from the observation should differ considerably, the operation must be repeated with the longitude obtained by this operation.

PROBLEM VI.

Given the latitude of the place, and the apparent times of the beginning and end of a solar eclipse, counted from noon to noon, according to the method of astronomers, to find the longitude of the place of observation.

In the rule for solving this problem, references will be made to figure 12, Plate XIII, in which DSE represents a small arc of the ecliptic; S, the place of the centre of the sur, supposed at rest; F, L, the apparent places of the centre of the moon at the beginning and end of the eclipse respectively; FD, SC, and AEL, are perpendicular to DE; FA parallel to DE, and SB perpendicular to FL. Then it is evident that FD, LE, represent the apparent latitudes of the moon, which fall below DE if south, above if north; and SF, SL, represent the sums of the corrected semi-diameters of the sun and moon, at the beginning and end of the eclipse respectively.

RULE.*

To the apparent times of the beginning and end of the eclipse, add the estimated longitude 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 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, and the remainder will be his corrected semi-diameter. Decrease the moon's semi-diameter 2" for inflexion, if it be thought necessary, and to the remainder add the correction in 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 altitudes and longitudes of the nonagesimal, by Problem IV.; 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 eclipse, and subtract therefrom the difference of the sun's longitudes at the same time; the remainder will be the relative motion in longitude DE or 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, the other south. From the logarithm 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 moon's apparent latitude FD, at the beginning of the eclipse, is greater than at the end EL, otherwise less.§ Then to the log.

This rule is peculiarly adapted to the use of the longitudes and latitudes of the bodies. We shall hereafter give the methods of performing the same calculations by means of the right ascensions and declinations, adapting the rules to the new form of the Nautical Almanac. The same is to be observed relative to the following Problems, VII. VIII., &c.

† 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.

This rule is equally true, whether the latitude be of the same or different names. If the latitudes are equal, and of the same name, the angle DSB will be 90°. If they are equal, but of different names, the angle DSB may be taken acute or obtuse, since, in that case, the angle FSB is 90°. Strictly speaking, when the points F, L, fall on different s'des of the line DE, the angle DSB is greater or less than 90°, according as the

cosecant of the angle of inclination, add the logarithm of the relative motion in longitude FA; the sum, rejecting 10 in the index, will be the logarithm of the apparent motion of the moon FL on her relative orbit. Then, in the triangle SFL, the sides SF, SL, represent the sums of the corrected semi-diameters of the sun and moon at the beginning and end of the eclipse, and these, with the relative motion FL, are given to find the angle FSB (by Case VI. Obl. 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 which, being added to and subtracted from 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 the 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. cosine of the central angle, add the logarithm of the sum of the corrected semidiameters at the beginning of the eclipse SF, rejecting 10 in the index; the sum will be the logarithm of SD, the apparent difference of longitude of the sun and moon at that time. This is to be subtracted from the longitude of the sun at the beginning of the eclipse, 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 eclipse.

Take the difference in seconds between the sun's and moon's true longitudes at the beginning of the eclipse, to the logarithm of which add the arith. comp. logarithm of the moon's horary motion from the sunt 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 eclipse, when the sun's longitude at that time is greater than the moon's true longitude, otherwise subtracted; the sum or difference will be the apparent time of the true ecliptic conjunction of the sun and moon at the place of observation. The difference between this and the time of conjunction at Greenwich, inferred from the Nautical Almanac by Problem III., will be the longitude of the place of observation. But if corresponding observations have been made at different places, it will be much more accurate to find the times of the conjunction at each place by the above rule; and the difference of these times will be 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 longitude and latitude may be easily found. For the difference between the moon's true longitude deduced by the above method from the observations, and the longitude found from the Nautical Almanac, will be the error of the tables in longitude. To find the error in latitude, add the log. sine of the central angle DSF to the logarithm of the sum of the corrected semi-diameters at the beginning of the eclipse SF; the sum, rejecting 10 in the index, will be the logarithm of the moon's apparent latitude FD at that time; which will be south, if the point F falls below D, otherwise north. Take the difference between this and the moon's apparent latitude, found by Problem V., if they are both north, or both south; but their sum, if one be north and the other south; and the error of the tables in latitude will be obtained.‡

REMARK.

The above rule will answer for deducing the longitude from the observed beginning and end of the internal contacts of a total or annular eclipse. The differences consist in reading

FD
SF

EL
SL

expression is greater or less than ; but, as the divisors SL and SF are nearly equal, they may be neg lected (as in the above rule), except in a case which very rarely occurs, namely, when the difference of SL, SF, is greater than the difference of the two apparent latitudes EL, FD, in which case the rule in this note EL FD must be made use of; observing that the fractions represent the quotients of the moon's apparent SL'SF latitudes divided by the sum of the semi-diameters of the sun and moon.

*When SF, SL, are equal, or their difference is so small that it may be neglected, the log. sine of the angle FSB may be obtained much more expeditiously by subtracting the logarithm of the sum of SF and SL from the logarithm of FL, increasing the index by 10. This method may almost always be made use of w 'the out much error. It is the rule adopted by Doctor Mackay in his treatise on longitude.

When the horary motion varies, it must be taken to correspond to the middle time between the beginning of the eclipse and the conjunction or new moon.

When the eclipse or occultation is nearly central, or (in other words) when FD, EL, are very small in comparison with SF, the latitude thus found cannot be depended on, as a small error in the times of observation will produce a considerable error in the latitude. Indeed, the case may occur, when FD), EL, are less than 30, that it may be uncertain whether the points F, L, fall above or below the line DE, because the error of the lunar tables in latitude may sometimes be equal to 30". In this case, the correct latitude of the moon may be found, (1.) By observations made at another place, where the eclipse or occultation was not so central; (2.) By the number of digits eclipsed, if it was a solar eclipse; (3.) By the difference of declinations of the moon and star, observed before and after the immersion or emersion; (4.) By the meridian altitude of the moon observed the same day, whence it may be found whether the moon was north or south of