2. What is the length of the longest day in the north of Spitzbergen, and on what days does it begin and

end?

3. What is the length of the longest day at the northern extremity of Nova Zembla ?

4. What is the length of the longest day at the north pole, and on what days does it begin and end?

PROBLEM XXXI.

To find the length of the longest night at any place in the north frigid zone.

Rule. Bring the given place to the northern point of the horizon (by elevating or depressing the pole,) and observe its distance from the north pole on the brass meridian; count the same number of degrees on the brass meridian from the equator towards the south pole, and mark the place where the reckoning ends; turn the globe on its axis, and observe what two points of the ecliptic pass under the above mark; find those points of the ecliptic in the circle of signs in the horizon, and exactly against them, in the circle of months, you will find the days on which the longest night begins and ends. The day preceding the 21st of December is that on which the longest night begins at the given place, and the day following the 21st of December is that on which the longest night ends: the space of time between these days is the length of the longest night.

OR, BY THE ANALEMMA.

Bring the given place to that part of the brass meridian which is numbered from the north pole towards the equator, and observe its distance in degrees from the pole; count the same number of degrees of the brass meridian from the equator towards the south pole, and mark where the reckoning ends; bring the analemma

*This problem is equally applicable to any place in the south frigid zone, and the rule will be general by reading south for north, and the contrary; likewise, instead of the 21st of December read the 21st of June.

to the brass meridian, and the two days which stand under the above mark will point out the beginning and end of the longest night.

Examples. 1. What is the length of the longest night at the North Cape, in the island of Maggeroe, in latitude 71° 30′ north?

Answer. The place is 183° from the pole; the longest night begins on the 16th of November, and ends on the 27th of January : the night is therefore seventy-three days long, that is, the sun does not rise during seventy-three revolutions of the earth on its

axis.

2. What is the length of the longest night at the north of Spitzbergen?

3. The Dutch wintered in Nova Zembla, latitude 76 degrees north, in the year 1596: on what day of the month did they lose sight of the sun; on what day of the month did he appear again; and how many days were they deprived of his appearance, setting aside the effect of refraction?

4. For how many days are the inhabitants of the northernmost extremity of Russia deprived of a sight of the sun?

PROBLEM XXXII.

To find the number of days which the sun rises and sets at any place in the north frigid zone.

Rule. Bring the given place to the northern point of the horizon (by elevating or depressing the pole,) and observe its distance from the north pole on the brass meridian; count the same number of degrees on the brass meridian, from the equator towards the poles northward and southward, and make marks where the reckoning ends; observe what two points of the ecliptic, nearest to Aries, pass under the above marks; these points will shew (upon the horizon) the end of the longest night and the beginning of the longest day; during the time between these days the sun will rise and set every twenty-four hours: next observe what two points of the ecliptic, nearest to Libra, pass under the marks on the brass meridian; find these points, as before, in the

*The same might be found for a place in the south frigid zone, were that zone inhabited.

circle of signs, and against them you will find the day on which the longest day ends at the given place, and the day on which the longest night begins; during the time between these days the sun will rise and set every twenty-four hours.

OR,

Find the length of the longest day at the given place (by Prob. XXX.) and the length of the longest night (by Problem XXXI.) add these together, and subtract the sum from 365 days, the length of the year, the remainder will shew the number of days which the sun rises and sets at that place.

OR, BY THE ANALEMMA.

Find how many degrees the given place is from the north pole, and mark those degrees upon the brass meridian on both sides of the equator; observe what four days on the analemma stand under the marks on the brass meridian; the time between those two days on the left hand part of the analemma (reckoning towards the north pole) will be the number of days on which the sun rises and sets, between the end of the longest night and the beginning of the longest day; and the time between the two days on the right hand part of the analemma (reckoning towards the south pole) will be the number of days on which the sun rises and sets, between the end of the longest day and the beginning of the longest night.

Examples. 1. How many days in the year does the sun rise and set at the north Cape, in the island of Maggeroe, in latitude 71° 30′ north?

Answer. The place is 181 from the pole, the two points in the ecliptic, nearest to Aries, which pass under 181° on the brass meridian are 8° in w, answering to the 27th of January, and 24o in 8, answering to 14th of May. Hence the sun rises and sets for 107 days, viz. from the end of the longest night, which happens on the 27th of January, to the beginning of the longest day, which happens on the 14th of May Secondly, the two points in the eclip tic, nearest to Libra, which pass under 183° on the brass meridian, are 8o in S, answering to the 30th of July, and 24o in m, answering to the 15th of November. Hence the sun rises and sets for 108 days, viz. from the end of the longest day, which happens on the 30th of July, to the beginning of the longest night, which happens on the 15th of November; so that the whole time of the sun's rising and setting is 215 days.

OR, THUS :

The length of the longest day, by Example 1st, Prob. XXX is 77 days, the length of the longest night, by Example 1st, Irob. XXXI. is 73 days; the sum of these is 150, which deducted from 365, leaves 215 days, as above.

2. How many days in the year does the sun rise and set at the north of Spitzbergen?

3. How many days does the sun rise and set at Greenland, in latitude 75° north?

4. How many days does the sun rise and set at the northern extremity of Russia in Asia?

PROBLEM XXXIII.

To find in what degree of north latitude, on any day between the 21st of March and the 21st of June, or in what degree of south latitude, on any day between the 23d of September and the 21st of December, the sun begins to shine constantly without setting; and also in what latitude in the opposite hemisphere he begins to be totally absent.

Rule. Find the sun's declination (by Prob. XX.) and count the same number of degrees from the north pole towards the equator, if the declination be north, or from the south pole, if it be south, and mark the point where the reckoning ends; turn the globe on its axis, and all places passing under this mark are those in which the sun begins to shine constantly without setting at that time: the same number of degrees from the contrary pole will point out all the places where twilight or total darkness begins.

Examples. 1. In what latitude north, and at what places does the sun begin to shine without setting during several revolutions of the earth on its axis, on the 14th of May ?

Answer. The sun's declination is 184° north, therefore all places in latitude 714° north will be the places sought, viz. the north Cape in Lapland, the southern part of Nova Zembla, Icy Cape, &c.

2. In what latitude south does the sun begin to shine without setting, on the 18th of October, and in what latitude north does he begin to be totally absent?

Answer The sun's declination is 10° south, therefore he begins to shine constantly in latitude 80° south, where there are no

inhabitants known, and to be totally absent in latitude 80° north, viz at Spitzbergen

3. In what latitude does the sun begin to shine without setting on the 20th of April?

4. In what latitude north does the sun begin to shine without setting on the 1st of June, and in what degree of south latitude does it begin to be totally absent ?

PROBLEM XXXIV.

Any number of days not exceeding 182, being given, to find the parallel of north latitude in which the sun does not set for that time.

Rule. Count half the number of days from the 21st of June on the horizon, eastward or westward, and opposite the last day you will find the sun's place in the circle of signs; look for the sign and the degree on the ecliptic, which bring to the brass meridian, and observe the sun's declination; reckon the same number of degrees from the north pole, (on that part of the brass meridian which is numbered from the equator towards the poles) and you will have the latitude sought.

Examples. 1. In what degree of north latitude, and at what places, does the sun continue above the horizon for seventy-seven days?

Answer Half the number of days is 381, and, if reckoned backward, or towards the east, from the 21st of June, will answer to the 14th of May; and if counted forward, or towards the west, will answer to the 30th of July; on either of which days the sun's declination is 18 degrees north, consequently the places sought are 18 degrees from the north pole, or in latitude 714 degrees north answering to the North Cape in Lapland, the south part of Nova Zembla, Icy Cape, &c

2. In what degree of north latitude is the longest day 134 days, or 3216 hours in length?

3. In what degree of north latitude does the sun continue above the horizon for 2160 hours?

4. In what degree of north latitude does the sun continue above the horizon for 1152 hours?