PROBLEM LV.

To find the time of the year when the sun or moon will be liable to be eclipsed.

Rule. 1. Find the place of the moon's nodes, the time of new moon, and the sun's longitude at that time, by an ephemeris ;* then, if the sun be within 17 degrees of the moon's node, there will be an eclipse of the sun.

2. Find the place of the moon's nodes, the time of full moon, and the sun's longitude at that time, by an ephemeris; then, if the sun's longitude be within 12 degrees of the moon's nodes, there will be an eclipse of the moon.

Examples. 1. On the 15th of January 1805, there was a full moon, at which time the place of the moon's nude was v 25° 54', and the sun's longitude 25°; did an eclipse of the moon happen at that time?

Answer. Here the sun was nearly in the moon's node, therefore a to al eclipse of the moon took pace; for, when the sun is in one of the moon's nodes at the time of full moon, the moon is in the other node, and the earth is directly between them; the moon's pl .ce was consequently ab ut 25° in Cancer.

2. It appears, by the foregoing table, that there was a new moon on the 30th of January 1805, at which time the place of the moon's node was 25° 16′, and the sun's longitude or place was m 10°; was there an eclipse of the sun at that time?

3. By the foregoing table, or by an ephemeris, there was a new moon on the 19th of October 1808, at which time the place of the moon's node was m 13o 6' and the sun's longitude 25° 56′; was there an eclipse of the sun at that time?

4 On the 3rd of November 1808, there will be a full moon, at which time the place of the moon's node. was m, 12° 18′ and the sun's longitude m 10° 55′ ; was there an eclipse of the moon at that time?

5. On the 4th of April 1810, there was a new moon, at which time the place of the moon's node

* White's Ephemeris, or the Nautical Almanac.

was 14° 57' and the sun's longitude was there an eclipse of the sun at that time?

14° 4′;

6. On the 28th of September 1810 there was new moon, at which time the place of the moon's node was 5° 32′ and the sun's longitude ≈ 4° 40′; was there an eclipse of the sun at that time?

PROBLEM LVI.

To explain the phenomenon of the harvest moon.

Definition 1. The harvest moon in north latitude, is the full moon which happens at, or near, the time of the autumnal equinox ; for to the inhabitants of north latitude, whenever the moon is in Pisces or Aries (and she is in these signs twelve times in a year,) there is very little difference between her ti nes of rising for sev eral nights together, because her orbit is at these times nearly parallel to the horizon. This peculiar rising of the moon passes unobserved at all other times of the year except in September and October; for there can never be a full moon except the sun be directly opposite to the moon; and as this particular rising of the moon can only happen when the moon is in Pisces or p Aries, the sun must necessarily be either in my Virgo or Libra at that time, and these signs answer to the months of September and October.

Definition.. The harvest moon, in south latitude, is the full moon which happens at, or near, the time of the vernal equinox; for, to the inhabitants of south latitude, whenever the moon is in my Virgo or Libra (and she is in these signs twelve times in a year) her orbit is nearly parallel to the horizon; but, when the full moon happens in my Virgo or Libra, the sun must be either in Pisces or p Aries. Hence it appears that the harvest moons are just as regular in south latitude as they are in north latitude, only they happen at contrary times of the year.

Rule for performing the problem.-1. For north latitude. Elevate the north pole to the latitude of the place, put a patch or make a mark in the ecliptic on the

point Aries, and upon every 12 * degrees preceding and following that point, till there be ten or eleven marks; bring that mark which is the nearest to Pisces to the eastern edge of the horizon, and set the index to 1; turn the globe westward till the other marks successively come to the horizon, and observe the hours passed over by the index; the intervals of time between the marks coming to the horizon will shew the diurnal difference of time between the moon's rising. If these marks be brought to the western edge of the horizon in the same manner, you will see the diúrnal difference of time between the moon's setting; for, when there is the smallest difference between the times of the moon's rising, there will be the greatest difference between the times of her setting; and, on the contrary, when there is the greatest difference between the times of the moon's rising, there will be the least difference between the times of her setting.

NOTE As the moon's nodes vary their position and form a complete revolution in about nineteen years, there will be a regular period of all the varieties which can happen in the rising and setting of the moon during that time. The following table (extracted from Ferguson's Astronomy) shews in what years the harvest moons are the least and most beneficial, with regard to the times of their rising, from 1805 to 1860. The columns of years under the letter L are those in which the harvest moons are least beneficial, because they fall about the descending node; and those under M are the most beneficial, because they fall about the ascending node.

* The reason why you mark every 12 degrees is, that the moon gains 12° 11' of the sun in the ecliptic every day (see the second note, page 46.)

† At London when the moon rises in the point Aries, the ecliptic at that point makes an angle of only 15 degrees with the horizon; but, when she sets in the pont Aries, it makes an angle of 62 degrees and, when the moon rises in the point Libra, the eclip tic, at that point, makes an angle of 62 degrees with the horizon; but, when she sets in the point Libra, it only makes an angle of 15 degree with the horizon.

2. For south latitude. Elevate the south pole to the latitude of the place, put a patch or make a mark on the ecliptic of the point Libra, and upon every twelve degrees preceding and following that point, till there be ten or eleven marks; bring that mark which is the nearest to Virgo, to the eastern edge of the horizon, and set the index to 12; turn the globe westward till the other marks successively come to the horizon, and observe the hours passed over by the index; the intervals of time between the marks coming to the horizon, will be the diurnal difference of time between the moon's rising, &c. as in the foregoing part of the problem.*

PROBLEM LVII.

The day and hour of an eclipse of any one of the satellites of Jupiter being given, to find upon the globe all those places where it will be visible.

Rule. Find the sun's declination for the given day, and elevate the pole to that declination; bring the place at which the hour is given to the brass meridian, and set the index of the hour circleste 12; then, if the given time be before noon, turn the globe westward as many hours as it wants of noon; if after noon, turn the globe eastward as many hours as it is past noon; fix the globe in this position: then,

1. If Jupiter rise after the sun†, that is, if he be an evening star, draw a line along the eastern edge of the horizon with a black lead pencil, this line will pass over all places on the earth where the sun is setting at the given hour; turn the globe westward on its axis till as many degrees of the equator have passed under the brass meridian as are equal to the difference between the sun's and Jupiter's right ascension; keep the globe

*This solution is on a supposition that the moon keeps constantly in the ecliptic, which is s fficiently accurate for illustrating the problem. Utherwise the latitude and longitude of the moon, or her right ascension and declination, may be taken from the Ephemeris, at the time of full moon, and a few days preceding and following it; her place will then be truly marked on the globe.

Jupiter rises after the sun, when his longitude is greater than the sun's longitude.

K k

from revolving on its axis, and elevate the pole as many degrees above the horizon as are equal to Jupiter's declination, then draw another line with a pencil along the eastern edge of the horizon: the eclipse will be visible to every place between these lines, viz. from the time of the sun's setting to the time of Jupiter's setting.

2. If Jupiter rise before the sun,* that is, if he be a morning star, draw a line along the western edge of the horizon with a black lead pencil, this line will pass over ali piaces of the earth where the sun is rising at the given hour; turn the globe eastward on its axis till as many degrees of the equator have passed under the brass meridian as are equal to the difference between the sun's and Jupiter's right ascension; keep the globe from revolving on its axis, and elevate the pole as many degrees above the horizon as are equal to Jupiter's declination, then draw another line with a pencil along the western edge of the horizon: the eclipse will be visible to every place between these lines, viz. from the time of Jupiter's rising to the time of the sun's rising.

Examples. 1. On the 13th of January 1805 there was an emersion of the first satellite of Jupiter at 9 m. 3 sec. past five o'clock in the morning, at Greenwich; where was it visible?

Answer. In this example the longitude of the sun exceeds the longitude of Jupiter; therefore Jupiter was a morning star, his declination being 19° 16' S. and his longitude 7 signs 29° 46', by the Nautical Almanac: his right ascension and the sun's right ascension may be found by the globe; for, if Jupiter's longitude in the ecliptic be brought to the brass meridian, his place will stand under the degree of his declination;† and his right ascension will be found on the equator, reckoning from Aries. This eclipse was visible at Greenwich, the greater part of Europe, the west of Africa, Cape Verd Islands, &c.

Jupiter rises before the sun, when his longitude is less than the sun's longitude.

This is on supposition that Jupiter moves on the ecliptic, and, as he eviates but little therefrom, the solution by this method will be sufficiently accurate. To know if an eclipse of any one of the sattelites of Jupiter will be visible at any place, we are directed by the Nautical Almanac, to "find whether Jupiter be 8° above the horizon of the place, and the sun as much below it."