latter longitude being brought to the meridian, the hourcircle will shew the time of her southing nearly.*

Ex. 1. At what time did the moon come to the meridian on the 25th of June, 1840, full moon having happened on the 15th of that month?

Here, since the moon is past full, and indeed in her last quarter, I find her longitude for the 24th or preceding day, and place a mark upon it :-Then, bringing the sun's place for the 25th of June to the brazen meridian, setting the hour-circle to 12, and giving the globe its westward motion until the patch coincides with the meridian, I find that nineteen hours are passed over by the hour-circle.-I therefore remove the patch eastward on the ecliptic, a quarter of a degree for each of these thirty-eight half-hours, or 94°, and find that the moon's longitude thus becoming 2° 8, she passes the meridian at nineteen hours and thirty-eight minutes from the last noon, or at seven hours thirty-eight minutes A.M. of June 25. (See "D's south." on p. 48.)

*** By referring to our page 48, I find that the exact time of the moon's southing was nine minutes later than this, or seven hours fortyseven minutes: this was owing to the quick motion of the moon in her orbit at that time (she being then near her perigee, as may be seen by her great horizontal parallax, in the last column on that page). If, instead of the above allowance of a quarter-degree for every halfhour, we had found the difference between her two longitudes, for the noons of the 24th and 25th, (viz., 14° 15′,) and had multiplied this difference by 19, and divided by 24, that is, had taken nineteen twenty-fourth parts of this, her rate of motion at that time, we should have found that she had shifted eastward not 91°, but 1110, during the nineteen hours; and this would have made our result coincide with the almanac time within two minutes. When she comes to the meridian only a few hours after noon, or when we learn from her parallax that she is not near her perigee, the allowance proposed in the rule will give an answer sufficiently correct.

Ex. 2. The moon was full on the 15th of June, 1840;


the moon

may suffice for illustration, since the average rate at which outstrips the sun in longitude is 12° 11' per twenty-four hours:-but for greater exactness, subtract the longitude of the moon for the preceding day from her longitude for the day given, and take so many twenty-fourths of this difference as are denoted by the number of hours, i. e., multiply this difference by the hours elapsed, and divide by 24; and it will shew the advance she has made eastward of her place at noon, and the longitude to be brought to the meridian. The change in the moon's latitude made during the hours from the last noon, will not materially affect the result sought for, and may be disregarded.

(see p. 48;) at what time did she come to the meridian of Greenwich on the 10th of June; her longitude for the several noons of that month being given on page 51?

Here we may safely use the allowance proposed in the rule; since we can tell by her parallax (on p. 48), that she was not near her perigee, and therefore not moving very fast at that time.

Ex. 3. The moon was full on the 5th of May, 1841:at what time did she "south" at Greenwich on the 14th of that month, her longitude for the noon of the 13th day being 20°


Ex. 4. The moon's longitude on 13th Aug,

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hor. paral.




She was full on the 2nd day of that month :- -at what time did she transit the meridian of Greenwich on the 14th of

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She was full on the 30th day :-at what time did she pass the meridian on the 20th day?

Ex. 6. 's long. Sept. 10, 1841, 12o 35' hor. par.

Sept. 11,


2' 了 60' She was full on the 1st day :-at what time will she pass the meridian on the 11th day?

Ex. 7. The moon was full on the 28th of November, 1841-at what hour did she " south" at Greenwich on the 5th of December?

her longitude on the 4th being 29° N 3'1

on the 5th

hor par. 58

13° m 6' f



From the time of the moon's southing on a certain day, and her declination for our noon of that day, to find where she will be vertical at that noon, nearly.

Admonitory Questions.

a If the moon and Spica (of Virgo) be together" southing," (or coming to our Greenwich meridian,) at 5 P.M.; in what east longi

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tude was the star southing, just two hours before, and in what longitude was the moon then southing, as she must then have been situate about 1° less east than that star?

b I observe the moon on my meridian at Greenwich at the same time with Antares, and just south of that star. The time is 5 A.M.:— in what west longitude will Antares be southing in three hours more; and in what longitude will the moon then be southing; since, in that time, she will have removed about 14° eastward of that star?

RULE.-If the southing be in the afternoon, turn the time elapsed from noon into degrees, and then subtract one quarter of a degree for every half-hour :* this will give the east longitude in which the moon is vertical at noon of that day.

If the southing be in the forenoon, subtract the given time from twelve hours; and the degrees answering to the difference of time, diminished by one quarter of a degree for every half-hour,* will shew the west longitude in which the moon is vertical at noon of that day. With the latitude answering to the declination given, and the longitude thus found, the place may be ascertained at which she was vertical (by Prob. XIV. Sect. 1.) `

Ex. 1. The moon's declination at noon was 15 south, and she was southing at Greenwich at half-past three, P.M.-where had she been vertical at noon?

Here 3 hours answering to 52° of longitude, the moon must have been situated that number of degrees to the east of our meridian, at the time of noon, if she had been fixed, like a star; but as her orbitual motion has been taking her eastward during those 3 hours, about 40 or 130, she must have been less eastward by that quantity at noon, and consequently vertical in 50° east longitude, which, in latitude 15° south, answers to Cape East, Madagascar.

Ex. 2. The moon, on a certain day, is culminating or southing at Blackheath, or London, at half-past six A.M.; her declination at noon will be 18 north-where will she then be vertical?

Ex. 3. D's declination for noon 22° 40′ :-she was half full, or just completing her first quarter, and passed my meridian at Blackheath at 6 P.M.; where must she have

* Because the average rate of the moon's eastward motion amongst the stars, is about half a degree per hour.

been vertical at our noon if she had been fixed, i. e., without orbitual motion; and where, from a consideration of her orbitual motion, was she then vertical?


If she had been fixed like a star, she must have been vertical in N. lat. 22° 40', and E. long. 90°; or • east of the zenith of but as, since our noon she had gone about 3o eastward, she evidently had been vertical in 3o less east than 90°, or • west of the zenith of that place.

Ex. 4. The moon is southing at London, or Greenwich, or Havre, at 12 minutes past 1 A.M.; where will her diminishing disk be shedding its lustre from the zenith, at the instant when the sun is on our meridian on this day: the moon's declination, as given for that instant in the Ephemeris, being 28° north?

Ex. 5. Fourteen days afterward, the moon, being about two days old, was southing 10 minutes past 2 P.M.; her declination at noon had been 28 south-where had her narrow crescent been discernible in the zenith at that instant ?

Ex. 6. On the 18th of May, in a certain year, the moon was eclipsed at 9 P.M. at London: her declination was 23° south (and she was over the heads of the people in the southern parts of Madagascar when thus eclipsed). She came to my meridian at Greenwich, at 15 minutes past 12, or midnight :—where will she be vertical next noon; her declination having, by that time, been changed from 23° to 203° south?

In the foregoing Rule, the allowance made for the moon's eastward motion, of a quarter of a degree for every half-hour, produces only an approximation to the truth; the moon's rate of motion varying with the time of year (p. 307), and with her distance from the earth (p. 303), from a little less than 12° to a little more than 15° per diem. When greater exactness is demanded, we may make use of the Rule of Three to compare the rate at which she is moving per 24 hours, with her quantity of motion for the time required. Thus, if from page 51, we wanted to find how much the moon moved in seven hours on the 2nd of June, 1840, we take her longitude of the 2nd of June, from her longitude of the third of June, and finding thus, her motion for those 24 hours to be 14° 41', we say

As 24 7: 14° 41' and we get 4° 17′ for 7 hours' motion.

In like manner, if the quantity of her motion for 7 hours of the 13th of June, 1840, were required, we find the difference of her longi. tude for the noons of the 13th and 14th, to be only 11o 53 and

As 24 7: 11° 53′: 3° 28' for the 7 hours of that day.




H Like that of every other heavenly body, the moon's stay above the horizon of any place depends upon the relation which her declination bears to its latitude.

We have learned (see quest. 6, 8, 12, p. 163) that at the North Pole, or in a Parallel Sphere, her continuance, like that of the sun, or of the rest of the planets, is for half her time at once, because an exact half of her course is north of the equinoctial, which there coincides with the horizon.

But the period of the moon's continuance above the horizon of all other places in the regions about the Pole, will be lengthened or shortened by the position of her node; for, as we have just seen, this affects the extent of her declination; and she cannot continue twenty-four hours without setting if her declination is less than the co-latitude. (See note preceding the Rule on p. 136.) When the moon's ascending node is in Aries, her declination (as we have seen from 7 of Prob. C) will become as much as 2830, and she may then continue more than the twenty-four hours even in as low a latitude as 6110*: whereas, if her ascending node be, on the contrary, in Libra, her declination cannot become more than 1810, and then many places even in the Frigid Zone will be without her continuous light; for this declination answers to the co-latitude of 7130, and it will be recollected that the polar circle is the parallel of 661°.

There are peculiarities in the moon's rising in our lower latitudes, which are occasioned by her rapid change of declination, and these are increased or diminished by the position of her node: but the pupil who has gone over Problem C, will be in no danger of being misled, if, for the sake of illustration, the ecliptic itself be taken as representing the average position of the moon's orbitual path.

At the equator, or in a Right Sphere, every degree of the eastward motion of the moon amongst the stars produces

* Little higher than the latitude of Petersburgh.

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