CONTAINING METHODS OF DETERMINING THE LONGITUDE BY OBSERVATIONS OF ECLIPSES, OCCULTATIONS, &c.

T

HE longitude of a place may be determined in a very accurate manner, by observing the beginning or end of a solar eclipse, or occultation of a fixed star by the moon, or the difference between the times that the moon and a known fixed star pass the meridian. These observations when made on land with a good telescope and well regulated time-keeper, furnish by far the most accurate method of determining the longitude, and when made ou board a ship without a telescope, will in general give it to a greater degree of accuracy than any other method. For this reason, it was thought proper to insert in this Appendix the usual rules of calculating such observations, by means of the Nautical Almanac. The first thing to be taken notice of, is the method of determining the longitude, latitude, &c. of the moon or other object, having regard to the unequal velocity between the times for which these quantities are given in the Nautical Almanac. This calculation is rendered much more simple by making use of the signs and, and performing addition and subtraction as in the introductory rules of Algebra; and as it is possible that these rules may not be familiar to some readers of this work, it was thought proper to prefix an explanation, as far as will be necessary in the present problems.

Quantities without a sign or with the sign+prefixed are called positive or affirmative, as 7 or +7; and those to which the sign is prefixed are called negative, as 7. Addition of quantities having the same sign, that is, all affirmative or all negative, is performed by adding them as in common arithmetic, and prefixing the common sign. Thus the sum of +4 and +3 is +7. The sum of 4, 3 and- 5 is

-

12. When the quantities have not the same sign, the positive quantities must be added into one sum, and the negative into another, as above; the difference of these two sums, with the sign of the greater sum prefixed, will be the sum of the proposed quantities. Thus the sum of + 14, 7, +5, and -2, is found by adding +14 and 5, whose sum is19; and then - - 7 and 2, whose sum is -9; the difference of 19 and 9 is 10, to which must be prefixed the sign of the greater number 19, which is +, so that the sought sum is The following examples will illustrate these rules. Add + Add +6' 0"

10.

Add + 4 Add 4' 10"
+2 5

+3 +7

- 2 Sum +6 15 Sum +12

Add 4' 10" Add 4' 10"
-2 5
+2 5
Sum-6 15 Sum-2 5

Sum 0

-2 15 +413 -3 V Sum +4 51

Subtraction is performed by changing the sign of the number to be subtracted from to, or from to +; and then adding the numbers by the preceding rule. Thus to subtract 3 from+7 the sign of +3 must be changed, and the numbers 3 and +7 added together as in algebra, which by the preceding rule gives + 4; and if it were required to subtract 3 from 7, the sign of - 3 must be changed, and +3, +7 added together. The sum + 10 represents the sought difference. It is not usual to make an actual change of the sign in any proposed question, it being sufficient to suppose the number to be subtracted to have a different sign from that prefixed to it, and to perform the operation accordingly. To illustrate this, the following examples are added.

Observing that when no sign is annexed to a quantity the sign is always under stood to be prefixed.

PROBLEM I.

To find the longitude, latitude, &c. of the moon at any given time at Greenwich, having regard to the unequal velocity between the times marked in the Nautical AlmaThe intervals of these times being 12 hours.

nac.

RULE.

Take from the Nautical Almanac the two longitudes, latitudes, &c. next preceding the given time at Greenwich, and the two immediately following it, and set them down in succession below each other, prefixing the sign to the southern latitudes or declinations, and the sign to the northeru. Subtract each of these quantities from the following for the first differences, and call the middle term arch A; subtract

M m (TAB.)

each first difference from the following for the second differences, and take the half sum or mean of them, which call the arch B, noting the signs of the quantities as in algebra.

Find the difference between the given time and the second time taken from the Nautical Almanac, which call T, then to its logarithm add the log. of A and the constant logarithm 5.36452, the sum rejecting 10 in the index, will be the logarithm of the proportional part, to which prefix the sign of the arch A; observing to express all these quantities in seconds.

Enter Table XLV. with the arch B at the top and the time T at the side,† opposite to this will be the correction of second differences, to which prefix a different sign from that of the arch B, and place it under the proportional part found above and the second quantity taken from the Nautical Almanac, and connect these three quantities together as in addition in algebra; the sum will be the sought longitude, latitude, &c. The latitude or declination being south if it has the sign+; north if it has the sign-‡ EXAMPLE I.

Required the longitudes and latitudes of the moon December 12, 1808, at 15h. 48' 29" and 17h. 1' 29" app. time by astronomical computation at Greenwich, which correspond to the immersion and emersion of Spica calculated in Problem VII.

Required the longitudes and latitudes of the moon June 16, 1906, at 2h. 49' 50.1 and Sh. 34 6.6 app. time, astronomical account at Greenwich, which correspond nearly to the beginning and end of the total eclipse of the sun as observed at Salem.

This may be foun to minutes by Table XXX. by entering it at the top with half the arch A (because the table extends only to 30 45) and at the side with the time T; the result doubled will be nearly the sought proportional part; but the table not being calculated to seconds it is hardly accurate enough to be used in calculating eclipses. This correction may also be found by proportion by saying as 12 hours are to the time T so is the arch A to the sought proportional part, and this method is the shortest when T is an aliquot part of 12 hours. Thus if T be 3, 6 or 9 hours, the proportional part will be or of the arch A respectively. This method is made use of in Problem XVII in interpointing the distances of the moon and sun.

If the arch B consists of minutes and seconds, the correction for minutes, tens of seconds and units of seconds must be found separately, the sum of these three parts will be the sought correction. Proportional parts for the minutes of the time T may be taken in finding the correction of this table, when necessary. In this role part of the correction of third differences is neglected. This part never exceeds via of the third differences, and rarely amounts to a small fraction of a second

In making use of the Nautical Almanac published by Mr. John Garnett, in which the polar distances for ours are given instead of the declination for 12 hours, the declination or polar distance

TO FIND THE LONGITUDE, &c. OF THE SUN, MOON, AND PLANETS. 557

The proportional parts of the arch A were calculated in this example by arithme. tic without logarithms. By observations of the eclipse on that day, it was found that the moon's longitude was too great by 58".5 and her latitude too great by 11".4. These corrections are applied to the above longitudes and latitudes in calculating the eclipse in Problem VI.

Remark 1. It will not be necessary to take notice of the second differences in calculating the parallax or semi-diameter of the moon, or any of the solar elements useful in calculating an eclipse or occultation. In this case the quantity immediately preceding and following the proposed time at Greenwich must be taken from the Nautical Almanac, and their difference will be the arch A, also the difference between the proposed time and that taken first from the Nautical Almanac is to be called the time T. Then by proportion, as the interval between the times taken from the Nautical Almanac is to the time T, so is the arch A to the correction to be applied to the first quantity taken from the Nautical Almanac, additive if increasing, subtractive if decreasing. This correction may also be found by logarithms as above, using the constant logarithms 5.36452 if the interval of the times in the Nautical Almanac is 12 hours, and 5.06349 if the interval is 24 hours. The proportional part of the moon's parallax and semi-diameter may also be found by Table XI. and that of the solar elements by Tables XXX. XXXI. as taught in the explanation of these tables. To exemplify this, the rest of the quantities requisite in calculating the eclipse and occultation (Prob. VI. VII.) are here found.

The semi-diameters thus found must be decreased 2" for inflexion, and augmented by the correction Table XLIV. in calculating an eclipse or occultation by Problem XIII. or in deducing the longitude from observations by Problems VI. VII. VIII. or IX.

The sun's semi-diameter by the Nautical Almanac, June 13, 1806, was 15′ 46′′.3 and June 19, 1806, was 15' 45'.9. Hence at the above time it was 15' 46".1. This . in eclipses of the sun must be decreased 34" for irradiation.

Remark 2. The above rule for calculating the second differences of the lunar motions where the intervals in the Nautical Almanac are 12 hours, may be made use of when the intervals are three, six, &c. days, (as is the case with the elements of the motions of the planets) by taking two longitudes, latitudes, &c. before, and two after, the given time at Greenwich, and thence deducing the arches A, B, and the longitudes, latitudes, &c. and then making use instead of T, of the quotient of the difference between the given time and that marked in the Nautical Alinanac against the second longitude, &c. divided by the number of half days in the given interval. Thus if the interval is one day, the divisor is two: if the interval is 3 days, the divisor is 6; and if the interval is 6 days, the divisor is 12. Thus if it were required to find the geocentric longitude of Jupiter July 14d. 13h. 30', 1808, astron. acc. at Greenwich, the work would be as follows.

In this example the time T, is 3h. 7' 30", found by dividing by 12, the interval be tween July 13 and July 14d. 13h. 30". In general the correction of Table XLV. may be neglected in calculating the places of the planets. In the above rule the intervals of time at which the longitudes, &c. are marked in the N. A. are supposed equal. If that should not be the case the correction Table XLV. may be neglected, on account of the trouble of calculating it.

PROBLEM II.

To find the horary motion of the moon in long. lat. &c. at any given time at Greenwich.

RULE.

Take from the Nautical Almanac the four longitudes, latitudes, &c. two immedi ately preceding the given time at Greenwich, and two immediately following. Prefix the sign to the southern latitudes or declinations, and the sign to the northThen find the first and second differences, the arch B, and the time T, as in Problem I. The mean of the two first differences, noticing the signs as in algebra, will be the approximate motion in 12 hours.

ern.

To the proportional logarithm of one fourth part of the time T add the proportional logarithm of the arch B, the sum will be the proportional logarithm of the correction of the approximate motion, to be applied to it with the same sign as the arch B, and the corrected motion of the moon in 12 hours will be obtained,* which divided by 12 will give the horary motion. EXAMPLE I.

Required the horary motions of the moon in longitude Dec 12, 1808, at 15h. 48' 29 and 17h. 1' 29' app. time at Greenwich.

This corresponds to Example I. preceding, in which T-3h. 48′ 29′′ and 5h. 1′ 29′′. The two first differences in longitude 7° 6′ 16′′ and 7° 11' 18'; their mean 7° 8′ 47′′ is the approximate motion in 12 hours, and the arch B is 4' 54".5. The rest of the calculation is as follows.

34' 21", and

-

In a similar manner if the Horary motion in latitude was required at 12d. 17h. 33' the two first differences in latitude are 36' 45', their mean - 35' 33" is the approximate motion in 12 hours. The correction found by the above rule with the time T. 5h. 33' and the arch B=-2′ 6′′.5 is 59' whence the true motion in 12 hours is - 36′ 32′′ which divided by 12 gives the horary motion-3′ 2.7. The negative sign indicates that the north polar distance is decreasing, the positive sign that it is increasing. In the present example the north polar distance was decreasing, and as the latitude was south it was also decreasing, as is evident.

EXAMPLE II.

Required the horary motious of the moon in longitude June 16, 1806, at 2h. 49' 50" 1 and 5h. 34' 6.6 app. time by astronomical computation at Greenwich.

This corresponds to Example II. preceding, in which T-2h. 49' 50.1 and 5h. 34' 6.6; the two first differences, are 7° 17′ 21′′ and 7° 20′ 53′′. the mean of which 7° 19' 7" is the approximate motion in 12 hours, the arch B is + 3′ 7′′.

*The motion in 12 hours thus obtained, which for distinction will be called the arch M, is not perfectly accurate since the third and higher orders of differences are neglected; but the horary motion deduced therefrom is abundantly sufficient for the purpose of projecting an eclipse or occultation. When greater accuracy is required, the third differences may be taken into account in the following fmanner. Having found the second differences as above directed, subtract the first of them from the second, noting the signs as in Algebra, and call the remainder the arch b. Enter Table XLV. with this arch at the top, and the time T at the side, and take out the corresponding correction which is to be increased by one sixth part of the arch b. without noting the signs. To the quantity thus found is to be prefixed a sign different from that of the arch b, and then it is to be applied to the arch M with its sign to obtain the true motion in 12 hours. Thus in the above example the second differences of longitude are 5' 2"+4' 47". Subtracting the former from the latter leaves the third difference or arch b=15". Corresponding to this and the time T 3h. 43' 29" in Table XLV. is 1.6. which increased by one sixth of b=2".5 gives the sought correction 4".1 or 4", to which must be preaxed the sign+(because the sign of b is negative) making it +4". This connected with the arch M+700' 20" gives the true motion in 12 hours 70 10 24" whence the horary motion is 35' 52". 15 a similar manner if the third differences were noticed in the above example for finding the horary motion in latitude, the two second differences are 2' 24" and 1'49" the arch b+35", the corFection of the motion in 12 hours 36′ 32" is -10" making it-36′ 42′′ or 3′ 3′′.5 per hour.

Mot. in 1 hour
EXAMPLE III.

36 42.8

Required the motion of the moon in right ascension in 12 hours, supposing it to increase uniformly with the velocity it had July 4th, 1808, at 9h. 26' app. time at Greenwich, by astronomical computation.

July 8, mid.

D. R. A
2250.57'

70 29' A 7 35

Approx. mot.
True mot. in 12h.

B 4' 30"
T-2h. 21 30
Correction + 3 32
7 32 0

7 35 32 or 30'.22".1 in time.

In this example T9h. 26′ and the approx. motion is the half sum of 7° 29' and 70 35'.

REMARKS.

1. When it is required to find the motion of the moon in any given interval of time, the motion in 12 hours must be found for the middle of that interval.

2. In calculating an occultation of a star by the moon, the relative borary motion in longitude is the same as the horary motion of the moon, because the star is at rest; but in calculating a solar eclipse, the sun's horary motion must be found in page 3 of the Nautical Almanac, and subtracted from the moon's horary motion in longitude, the remainder will be the horary motion of the moon from the sun in longitude. Thus on the 16th of June, 1806, the sun's horary motion was 2′ 23′′.1, which subtracted from the horary motions found in Example II. 36'.39.2 and 36′ 42′′.8 leaves the corresponding horary motions of the moon from the sun in longitude 34' 16".1 and 34′ 19′′. 7.

As the sun has no sensible motion in latitude, the relative horary motion of the moon from the sun in latitude, is the same as the true horary motion of the moon in latitude.

half sum of the two first B=-7 P. L. 1.4102 T46 52 P. L.

3. The horary motion of a planet may be found in a similar manner, making use of the arches A. B. T. found as in Remark 2, Problem I. Thus if the horary motion of Jupiter was required July 14, 1808, at 13h. 30', the arch B=-7' T=3h. 7′ 30′′, and the approximate motion in the interval 6 days is the differences-3′ and-11', namely-7′ 0′′. The correction found as in the adjoining calculation is-1' 49", hence the motion in 6 days is-8' 49", whence the horary motion is-3", 67. The negative sign indi Corr. cates that the motion is retrograde, or contrary to the order of the signs: in this case the relative notion of Ap. ino.-7 the moon from the planet in longitude would be found

-

5844

1 49 P. L. 1.9946

0

by adding their horary motions, because the motion (Motion - 49 in six days. of the moon is always direct. Similar remarks may be made in finding the horary motion of the moon from the planet in latitude.

PROBLEM JII.

To find the time of the ecliptic conjunction or opposition of the moon with the sun, a planet, or a fixed star.

The time of the ecliptic conjunction of the sun and moon is the same as the time of new moon given for the meridian of Greenwich in the first page of the month of the Nautical Almanac. Thus in January 1808, the ecliptic conjunction is on the 27th day at 4h. 9' apparent time at Greenwich. The times of the ecliptic conjunction of the moon and those fixed stars with which there may be an occultation are also given in the same page, being marked with Bayer's characters of reference. The time of conjunction is placed first, then the characters of the moon and star, or moon and planet. Thus in 1808, December 12d. 17h. 33' a, signifies that on the 12th day of December at 17h. 33' apparent time at Greenwich, the moon was in ecliptic conjunction with the star Spica, whose character is a M, and that there might be an occultation of that star. Also December 15, 1808, 5h. 53') signifies that at that moment apparent time at Greenwich, the moon and Saturn were in ecliptic conjunction, and there might be an occultation of that planet. These times being reckoned according to astronomical computation, and in calculating them no attention is paid to the parallaxes. The time of the ecliptic opposition of the sun and moon is the same as at the time of full moon given in the same page of the Nautical Almanac. Thus the full moon or ecliptic opposition in May, 1803, was 9d. 13h. 39′ at Greenwich: