through the east and west points thereof, reckoning towards the north, are in darkness, notwithstanding they are above the horizon; and all places below the horizon, between those same meridians and the southern point of the horizon, have day-light notwithstanding they are below the horizon of London.

PROBLEM XXIII.

The month and day of the month being given, to find all places of the earth where the sun is vertical on that day; those places where the sun does not set, and those places where he does not rise on the given day.

Rule. Find the sun's declination (by ProblemXX.) for the given day, and mark it on the brass meridian; turn the globe round on its axis from west to east, and all the places which pass under this mark will have the sun vertical on that day.

Secondly. Elevate the north or south pole, according as the sun's declination is north or south, so many degrees above the horizon as are equal to the sun's declination; turn the globe on its axis from west to east ; then, to those places which do not descend below the horizon, in that frigid zone near the elevated pole, the sun does not set on the given day and to those places which do not ascend above the horizon, in that frigid zone adjoining to the depressed pole, the sun does not rise on the given day.

OR, BY THE ANALEMMA.

Bring the analemma to that part of the brass meridian which is numbered from the equator towards the poles, the degree directly above the day of the month, on the brass meridian, is the sun's declination. Elevate the north or south pole, according as the sun's declination is north or south, so many degrees above the horizon as are equal to the sun's declination; turn the globe on its axis from west to east, then to those places which pass under the sun's declination on the brass meridian, the sun will be vertical; to those places (in that

frigid zone near the elevated pole) which do not go below the horizon, the sun does not set; and to those places (in that frigid zone near the depressed pole) which do not come above the horizon, the sun does not rise on the given day.

Examples. 1. Find all the places of the earth where the sun is vertical on the 11th of May, those places in the north frigid zone where the sun does not set, and those places in the south frigid zone where he does not rise.

Answer. The sun is vertical at St. Anthony, one of the Cape Verd islands, the Virgin Islands, south of St. Domingo, Jamaica, Golconda, &c. All places within eighteen degrees of the north pole will have constant day; and those (if any) within eighteen degrees of the south pole will have constant night.

2. Does the sun shine over the north or south pole on the 27th of October? To what places will he be vertical at noon? What inhabitants of the earth will have the sun below their horizon during several revolutions, and to what part of the globe will the sun never set on that day?

3. Find all the places of the earth where the inhabitants have no shadow when the sun is on their meridian, on the first of June?

4. What inhabitants of the earth have their shadows directed to every point of the compass, during a revolution of the earth on its axis, on the 15th of July?

5. How far does the sun shine over the south pole on the 14th of November? What places in the north frigid zone are in perpetual darkness? And to what places is the sun vertical?

6. If the sun be vertical at any place on the 15th of April, how many days will elapse before he is vertical a second time at that place?

7. If the sun be vertical at any place on the 20th of August, how many days will elapse before he is vertical a second time at that place?

8. Find all places of the earth where the moon will be vertical on the 15th of May, 1813.*

*To perform this example, find the moon's declination on the given day in the Nautical Almanac, or White's Ephemeris, and mark it on the brass meridian, all places passing under that degree of declination will have the moon vertical, or nearly so, on the given day. The moon's declination at midnight on the 15th of May, 1813, is 15° 46′ south.

PROBLEM XXIV.

A place being given in the torrid zone, to find those two days of the year on which the sun will be vertical at that place.

Rule. Bring the given place to that part of the brass meridian which is numbered from the equator towards the poles, and mark its latitude; turn the globe on its axis, and observe what two points of the ecliptic pass under that latitude; seek those points of the ecliptic in the circle of signs, on the horizon, and exactly against them, in the circle of months, stand the days required.

OR, BY THE ANALEMMA.

Find the latitude of the given place (by Problem I.) and mark it on the brass meridian; bring the analemma to the brass meridian, upon which, exactly under the latitude, will be found the two days required.

Examples. 1. On what two days of the year will the sun be vertical at Madras ?

Answer. On the 25th of April and on the 18th of August.

2. On what two days of the year is the sun vertical at the following places?

The month and the day of the month being given (at any place not in the frigid zones,) to find what other day of the year is of the same length.

Rule. Find the sun's place in the ecliptic for the given day (by Problem XX.) bring it to the brass meridian and observe the degree above it; turn the globe on its axis till some other point of the ecliptic falls under the same degree of the meridian; find this point of

the ecliptic on the horizon, and directly against it you will find the day of the month required.

This problem may be performed by the celestial globe in the same

manner.

OR, BY THE ANALEMMA.

Look for the given day of the month on the analemma, and adjoining to it you will find the required day of the month.

OR, WITHOUT A GLOBE.

Any two days of the year which are of the same length, will be an equal number of days from the longest or shortest day. Hence, whatever number of days the given day is before the longest or shortest day, just so many days will the required day be after the longest or shortest day, et contra.

Examples. 1. What day of the year is of the same length as the 25th of April?

Answer. The 18th of August.

2. What day of the year is of the same length as the 25th of May?

3. If the sun rise at four o'clock in the morning at London on the 17th of July, on what other day of the year will he rise at the same hour?

4. If the sun set at seven o'clock in the evening at London on the 24th of August, on what other day of the will he set at the same hour?

year

5. If the sun's meridian altitude be 90° at Trincomalé, in the island of Ceylon, on the 12th of April, on what other day of the year will the meridian altitude be the same?

6. If the sun's meridian altitude at London, on the 25th of April, be 51° 35′, on what other day of the year will the meridian altitude be the same?

PROBLEM XXVI.

The month, day, and hour of the day being given, to find where the sun is vertical at that instant.

Rule. Find the sun's declination (by Problem XX.) and mark it on the brass meridian; bring the given

place to the brass meridian, and set the index of the hour circle to twelve; then, if the given time be before noon, turn the globe westward as many hours as it wants of noon; but, if the given time be past noon, turn the globe eastward as many hours as the time is past noon; the place exactly under the degree of the sun's declination will be that sought.

Examples. 1. When it is forty minutes past six o'clock in the morning at London on the 25th of April, where is the sun vertical?

Answer. Here the given time is five hours twenty minutes before noon; hence, the globe must be turned towards the west till the index bas passed over five hours and twenty minutes,* and under the sun's declination, on the brass meridian, you will find Madras, the place required.

2. When it is four o'clock in the afternoon at London, on the 18th of August, where is the sun vertical?

Answer. Here the given time is four hours past noon; hence, the globe must be turned towards the east, till the index has passed over four hours; then under the sun's declination, you will find Barbadoes, the place required.

3. When it is three o'clock in the afternoon at Lon. don, on the 4th of January, where is the sun vertical?

4. When it is three o'clock in the morning at London, on the 11th of April, where is the sun vertical?

5. When it is thirty-seven minutes past one o'clock in the afternoon at the Cape of Good Hope, on the 5th of February, where is the sun vertical?

6. When it is eleven minutes past one o'clock in the afternoon at London, on the 29th of April, where is the sun vertical?

7. When it is twenty minutes past five o'clock in the afternoon at Philadelphia, on the 18th of May, where is the sun vertical?

8. When it is nine o'clock in the morning at Calcutta, on the 11th of April, where is the sun vertical?

* If the hour circle be not divided to twenty minutes, turn the globe westward till the index has passed over five hours and a quarter; then by turning it a degree and a quarter farther to the west (answering to five minutes of time) the solution will be exact. See the note to the next problem. The degrees must be counted on the equator.

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