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, Prob. 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. Now 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 183 north, therefore, all places in latitude 713° 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 lati tude 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 op. posite to the last day you will find the sun's place in the circle of signs; look for the sign and 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 384, 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?

PROBLEM XXXV.

To find the beginning, end, and duration of twilight at any place, on any given day.

Rule. Find the sun's declination for the given day (by Problem XX.) and elevate the north or south pole, according as the declination is north or south, so many degrees above the horizon as are equal to the sun's declination; screw the quadrant of altitude on the brass meridian, over the degree of the sun's declination; bring the given place to the brass meridian, and set the index of the hour circle to twelve ; turn the globe eastward till the given place comes to the horizon, and the hours passed over by the index will show the time of the sun's setting, or the beginning of evening twilight; continue the motion of the globe eastward, till the given place coincides with 18° on the quadrant of altitude below* the horizon, the space passed over by the index of the hour circle from the time of the sun's setting, will show the duration of evening twilight. The morning twilight is the same length.

OR, THUS :

Elevate the north or south pole, according as the latitude of the given place is north or south, so many degrees above the horizon as are equal to the latitude; find the sun's place in the ecliptic, bring it to the brass meridian, set the index of the hour circle to twelve, and screw the quadrant of altitude upon the brass meridian over the given latitude; turn the globe westward on its axis till the sun's place comes to the western edge of the horizon, and the hours passed over by the index will show the time of the sun's setting, or the beginning of evening twilight; continue the motion of the globe westward till the sun's place coincides with 18° on the quadrant of altitude below the horizon, the space passed over by the index of the hour circle, from the time of the sun's setting, will show the duration of evening twilight.

The quadrant of altitude belonging to our modern globes is always graduated to 18 degrees below the horizon.

OR, BY THE ANALEMMA.

Elevate the pole to the latitude of the place, as above, and screw the quadrant of altitude upon the brass meridian over the degree of latitude; bring the middle of the analemma to the brass meridian, and set the index of the hour circle to twelve: turn the globe westward till the given day of the month on the analemma comes to the western edge of the horizon, and the hours passed over by the index will show the time of the sun's setting, or ' the beginning of evening twilight: continue the motion of the globe westward till the given day of the month coincides with 18° on the quadrant below the horizon, the space passed over by the index, from the time of the sun's setting, will show the duration of evening twilight.

Examples. 1. Required the beginning, end, and duration of morning and evening twilight at London, on the 19th of April.

Answer. The sun sets at two minutes past seven, and evening twilight ends at nineteen minutes past nine: consequently, morning twi light begins at (12 h.-9 h. 19 m.) 2 h. 41 m. and ends at (12 h.-. 7 h.2 m. =) 4 h. 58 m.; the duration of twilight is 2 hours 17 minutes.

2. What is the duration of twilight at London on the 23d of September? what time does dark night begin? and at what time does day break in the morning?

Answer. The sun sets at six o'clock, and the duration of twilight is two hours; consequently, the evening twilight ends at eight o'clock, and the morning twilight begins at four.

3. Required the beginning, end, and duration of morning and evening twilight at London, on the 25th of August.

4. Required the beginning, end, and duration of morning and evening twilight at Edinburgh, on the 20th of February.

5. Required the beginning, end, and duration of morning and evening twilight at Cape Horn, on the 20th of February.

6. Required the beginning, end, and duration of morning and evening twilight at Madras, on the 15th of June.

PROBLEM XXXVI.

To find the beginning, end, and duration of constant day or twilight at any place.

Rule. Find the latitude of the given place, and add 18° to that latitude; count the number of degrees correspondent to the sum, on that part of the brass meridian which is numbered from the pole towards the equator, mark where the reckoning ends, and observe what two points of the ecliptic pass under the mark ;* that point wherein the sun's declination is increasing will show on the horizon the beginning of constant twilight; and that point wherein the sun's declination is decreasing, will show the end of constant twilight.

Examples. 1. When do we begin to have constant day or twilight at London, and how long does it continue ?

Answer. The latitude of London is 51 degrees north, to which add 18 degrees, the sum is 694, the two points of the ecliptic which pass under 694 are two degrees in I, answering to the 22d of May; and 29 degrees in, answering to the 21st of July; so that, from the 22d of May to the 21st of July, the sun never descends 18 degress below the horizon of London.

2. When do the inhabitants of the Shetland islands cease to have constant day or twilight?

3. Can twilight ever continue from sun-set to sunrise at Madrid?

4. When does constant day or twilight begin at Spitzbergen?

5. What is the duration of constant day or twilight at the North Cape in Lapland, and on what day, after their long winter's night, does the sun's rays first enter the atmosphere?

* If, after 18 degrees be.added to the latitude, the distance from the pole will not reach the eliptic, there will be no constant twilight at the given place: viz. to the given latitude add 13 degrees, and subtract the sum from 90, if the remainder exceed 234 degrees, there can be no constant twilight at the given place.