diurnal path in the heavens will be along that line, and he will be vertical to all the inhabitants on the earth in latitude 23° 28' north. The inhabitants upon the equa tor will always have twelve hours day and twelve hours night, notwithstanding the variation of the sun's declination from north to south, or from south to north because the parallel of latitude which the sun apparently describes for any day will always be cut into two equal parts by the horizon. The greatest meridian altitude of the sun will be 90°, and the least 66° 32′• Duig one half of the year, an inhabitant on the equator will see the sun full north at noon, and during the other half it will be fuil south.

2. For the Parallel Sphere. The inhabitants (if any) who live at the north pole, have a parallel sphere and the north polar star in the heavens appears exactly (or very nearly,) over their heads. Elevate the north pole ninety degrees above the horizon, then the equator will coincide with the horizon, and all the parallels of latitude will be parallel thereto. In the summer half year, that is, from the vernal to the autumnal equinox, the sun will appear above the horizon, consequently the stars and planets will be invisible during that period, When the sun enters Aries, on the 21st of March, he will be seen by the inhabitants of the north pole (if there be any inhabitants) to skim just along the edge of the horizon; and, as he increases in declination, he will increase in altitude, forming a kind of spiral, as before described, by wrapping a thread round the globe. The sun's altitude at any particular hour is always equal to his declination. The greatest altitude the sun can have is 23° 28', at which time he has arrived at the tropic of Cancer; after which he will gradually decrease in altitude as his declination decreases. When the sun arrives at the sign Libra, he will again appear› to skim along the edge of the horizon, after which he will totally disappear, having been above the horizon for six months. Though the inhabitants at the north pole will lose sight of the sun a short time after the autumnal equinox, yet the twilight will continue nearly two months; for the sun will not be 18° below the horizon till he enters the 20th of Scorpio, as may be seen by the globe.

After the sun has descended 18° below the horizon, all the stars in the northern hemisphere will become visible, and appear to have a diurnai revolution round the earth from east to west, as the sun appeared to have when he was above the horizon. These stars will not set during the winter half of the year; and the planets, when they are in any of the northern signs, will be visible. The inhabitants under the north polar star have the moon constantly above their horizon during fourteen revolutions of the earth on its axis, and at every full moon which happens, from the 23d of September to the 21st of March, the moon is in some of the northern signs, and consequently visible at the north pole; for the sun being below the horizon at that time, the moon must be above the horizon, because she is always in that sign which is diametrically opposite to the sun at the time of full moon.

When the sun is at his greatest depression below the horizon, being then in Capricorn, the moon is at her first quarter in Aries: full in Cancer; and at her third quarter in Libra: and, as the beginning of Aries is the rising point of the ecliptic, Cancer the highest, and Libra the setting point, the moon rises at her first quarter in Aries, is most elevated above the horizon and full in Cancer, and sets at the beginning of Libra in her third quarter; having been visible for fourteen revolutions of the earth on its axis, viz. during the moon's passage from Aries to Libra. Thus the north pole is supplied one half of the winter time with constant moonlight in the sun's absence; and the inhabitants only lose sight of the moon from her third to her first quarter, while she gives but little light, and can be of little or no service to them.

3. For the Oblique Sphere.-Whenever the terrestrial globe is placed in a proper situation with respet to the the fixed stars, the pole must be elevated as many degrees above the horizon as are equal to the latitude of the given place, and the north pole of the glote must point to the north polar star in the heavens; for in sailing, or travelling from the equator northward, the north polar star appears to rise higher and higher. On the equator it will appear in the horizon; in ten degrees of north latitude it will be ten degrees above the hori

zon; in twenty degrees of north latitude it will be twenty degrees above the horizon; and so on, always increasing in altitude as the latitude increas

es.

Every inhabitant of the earth, except those who live upon the equator, or exactly under the north polar star, has an oblique sphere, viz. the equator cuts the horizon obliquely. By elevating and depressing the poles, in several problems, a young stu dent is sometimes led to imagine that the earth's axis moves northward and southward, just as the pole is raised or depressed: this is a mistake; the earth's axis has no such motion.* In travelling from the equator northward, our horizon varies: thus, when we are on the equator, the northern point of our horizon is exactly opposite the north polar star; when we have trayelled to ten degrees north latitude, the north point of our horizon is ten degrees below the pole, and so on: now, the wooden horizon on the terrestrial globe is immoveable, otherwise it ought to be elevated or depressed, and not the pole ; but whether we elevate the pole ten degrees above the horizon, or depress the north point of the horizon ten degress below the pole, the appearance will be exactly the same.

The latitude of London is about 51 degrees north: if London be brought to the brass meridian, and the north pole be elevated 514 degrees above the north point of the wooden horizon, then the wooden horizon will be the true horizon of London; and, if the artificial globe be placed exactly north and south by a mariner's compass, or by a meridian line, it will have exactly the position which the real globe has. Now, if we imagine lines to be drawn through every degreet within the torrid zone, parallel to the equator, they will nearly represent the sun's diurnal path on any given day. By comparing these diurnal paths with each other, they will be found to increase in length from the equator northward, and to decrease in length from the equator south

* The earth's axis has a kind of librating motion, called the nutation, but this cannot be represented by elevating or depressing the pole. Such lines are drawn on Adams' globes.

ward; consequently, when the sun is north of the equator, the days are increasing in length, and when south of the equator, the days are decreasing. The sun's meridian altitude for any day may be found by counting the number of degrees from the parallel in which the sun is on that day, towards the horizon, upon the brass meridian; thus, when the sun is in that parallel of latitude which is ten degrees north of the equator, his meridian altitude will be 481 degrees. Though the wooden horizon be the true horizon of the given place, yet it does not separate the enlightened hemisphere of the globe from the dark hemisphere, when the pole is thus elevated. For instance, when the sun is in Aries, and London at the meridian, all the places on the globe above the horizon beyond those meridians which pass 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 Problem XX.) 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 declinaton; 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 Hear 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 meri dian 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 poie, 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 velow 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 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 to St Anthony, one of the Cape Verd islands, the Virgin islands, south of St. Domingo, Jamaica, Golconda, &c. Ali 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. Whether 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 nave the sun beiow 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 perpetuai darkness? and to what places is the sun vertical?

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