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vertical twice a year ? At what hour of the day is the sun vertical to any place ? How often in the year is the sun vertical to places in the north temperate zone? How often is the sun vertical at the tropics ? To what tropic is the sun vertical on June 21st? At what tropic is the sun vertical on December 21st ? Why is the sun never vertical to places that are not in the torrid zone? How is it found on what two days of the year the sun is vertical to any place in the torrid zone? How is this found without the globe? How may the examples be proved ? How are the places found to which the sun is vertical on any given day? Having the day and hour given, how is it found where the sun is then vertical ? A place being given in the north frigid zone, how is it found when the sun begins to appear above the horizon, and when it begins to disappear; also, the length of the longest day and night? On any given day between the vernal equinox and summer solstice, how is it found in what latitude in the north frigid zone the sun begins to shine without setting ?

QUESTIONS FOR EXERCISE IN SECTION V. 1. To what places will the sun be vertical on April 9th ? 2. On March 12th ? 3. On August 15th ? 4. When will the sun pass vertically over Surinam ?

5. When will the sun pass vertically over the islands of Ascension, Mauritius and Guam ?

6. On what two days in the year will a person at St. Christopher's have no shadow at noon ?

7. To what place will the sun be vertical on January 31st, when it is 9 in the morning at Newcastle ?

8. On June 14th, when it is 3-past 8 in the morning at Newcastle, where is the sun vertical ?

To what places will the sun be vertical -
9. On July 12th, when it is 9 in the evening at Jerusalem ?
10. On the 26th of June, when it is 7 a.m. at London ?
11. On July 10th, when it is 2 in the morning at Quebec ?
12. On Christmas-day, when it is midnight at Petersburg ?
13. On November 10th, when it is 6 in the evening at Ormus ?
14. Is the sun ever vertical at Jerusalem ?

15. Is the sun ever more than 24 hours above the horizon at Archangel ?

16. Suppose a person to pass the winter in 77° N. latitude, how long would he be without seeing the sun ?

17. In what latitude does the sun begin to shine, without setting, on May 1st ?

SECTION VI.

DEFINITIONS. 1. Twilight is that medium between light and darkness which happens before sunrise, and after sunset.

2. The crepusculum is a circle parallel to the horizon, and 18° below it, where the twilight begins and ends.

Of Twilight.-As soon as the sun comes within 18° of the horizon, its rays strike the higher parts of the atmosphere, and, being refracted and reflected to every part, occasion that agreeably gradual transition from darkness to light, called twilight.

In the same manner in the evening, after the sun sets, its rays strike upon the higher parts of the atmosphere, until it is more than 18° below the horizon : this prevents us from being suddenly deprived of the light of the sun.

The benefits of twilight are obvious. A change so great, as from the darkness of midnight to the splendour of noon-day, would probably be injurious to the sight; and it would be unpleasant to all, and in many cases very dangerous to travellers, to be involved in darkness without timely notice of its approach.

In countries near the equator, twilight is of much shorter duration than it is in countries of high latitudes; for at the equator the sun rises and sets perpendicularly, and consequently the twilight there cannot be greater than 1 hour 12 minutes,—but to places at a great distance from the equator it rises and sets very obliquely; and hence it requires a longer time to go 18° below the horizon.

At the latitude of 49° N. twilight continues the whole night on June 21st; and, at places still farther north, it continues the whole night, for a certain number of days before and after the summer solstice. At London there is no total darkness from May 28th till July 20th.

Twilight continues, at the north pole, from September 23rd, when the sun sets, to November 12th,—a space of 51 days. Twilight first appears again there about the 30th of January, and continues till sunrise on March 21st. Thus, though the inhabitants (if any) at the north pole never see the sun for 6 months, yet, out of that time, they have twilight for 14 weeks. The time that they receive no light from the sun is only 12 weeks; and during that time the moon is 6 weeks above the horizon.

PROBLEM XXX. To find at whal place it is Twilight at any given time.

Find where the sun is then vertical, and elevate the globe for that place. Observe what places are less than 18° below the horizon; to those below the western semicircle it is twilight in the morning, and to those below the eastern semicircle it is twilight in the evening.

Otherwise.— Elevate the globe for the antipodes of the place to which the sun is then vertical, and observe what places are within less than 18" above the horizon.

EXAMPLES.

1. On March 10th, when it is 11 p.m. at New Orleans, where is it twilight ?

Ans. Morning.-Britain, France, middle of Africa. Even. Twilight.-Society and Sandwich Isles, Alaska.

2. When it is 6 hrs. 45 min. a.m. at Newcastle on April 27th, where is it twilight? Ans. Morning-Labrador, Newfoundland. Evening—Alaska, New Hebrides, New Caledonia, part of New Zealand.

3. Where is it twilight, when it is 3 o'clock p.m. at London, on June 4th ?

4. On September 25th, when it is 10 p.m. at Trinidad ? 5. Dec. 16th, when it is noon at Easter Island ? 6. On April 4th, when it is 6 a.m. at Edinburgh ? 7. On June 9th, when it is 3 a.m. at Glasgow ? 8. On March 24th, when it is 4 p.m. at Benares ?

PROBLEM XXXI. To find the Duration of Twilight on a given day. 1. Elevate the globe for the latitude of the place, bring the sun's place to the meridian, and set the index to 12.

2. Turn the globe till the sun's place be 18° below the horizon, and the index will show the beginning of twilight; or that point in the ecliptic, opposite to the sun's place, may be brought 18° above the western horizon.

3. Subtract the commencement of twilight from the time of the sun's rising, (see Problem XVII.,) and the remainder will be the duration of twilight.

EXAMPLES.

How long does twilight continue at London on-
1. March 2d ?

Ans. 1 hr. 50 min.
2. June 21st ?

Ans. No night.
3. September 25th ? Ans. 2 hrs. O min.

4. December 26th ? Ans. 2 10
How long does twilight continue on March 21st at-

5. Cape of Good Hope ? 8. Vienna ?
6. Quito ?

9. Petersburg ?
7. Jerusalem ?
How long will it continue on June 21st at-
10. Cape Horn ?

13. Pekin?
11. Gilolo Isle ?

14. St. Petersburg ? 12. Formosa Isle ?

15. Cairo ? PROBLEM XXXII. To find at what places an Eclipse of the Moon is visible.

Find, by Problem XXIV., the place to which the sun is vertical at the given time. Elevate the globe for the antipodes of that place, and bring the antipodes to the meridian: then, to all the places which are above the horizon, the eclipse will be visible.

At an eclipse of the moon, the sun and moon are in opposite points of the ecliptic; and the place to which the moon is then vertical is the antipodes of that to which the sun is vertical.

EXAMPLES. 1. On May 10th, 1808, there was a total eclipse of the moon when it was 8 o'clock in the morning at Greenwich; where was it visible? Ans. N. and S. America, the islands in the Pacific Ocean, east coast of New Holland.

2. On April 30th, 1809, there was an eclipse of the sun when it was 1 am, at London ; where was it visible?

3. In 1811, March 10th, there was an eclipse of the moon, at 61 a.m. at London ; where was it visible ?

4. Aug. 22nd, 1812, there was an eclipse of the moon when it was 3 p.m. at London ; where was it visible ?

QUESTIONS FOR EXAMINATION IN SECTION VI. What is it that produces twilight? In what countries is twilight of the shortest duration? In what latitude does twilight continue the whole night on June 21st ? How long does twilight continue the whole night at London, and places of the same latitude as London? How is it found where it is twilight at any given time? How is the duration of twilight found at any place on any given day? How is it found where an eclipse of the moon is visible?

PART III.

CONTAINING PROBLEMS ON THE CELESTIAL

GLOBE.

DEFINITIONS. 1. The celestial globe is an artificial representation of the heavens, having the fixed stars delineated upon it, in their natural order and situation.

The celestial globe is not so exact a representation of the heavens as the terrestrial globe is of the earth; because the stars are drawn upon a convex surface, and they appear in the heavens in an inverted order on a concave surface: but suppose the globe were made of glass, then, to an eye placed in the centre, the stars drawn upon it would appear on a concave surface, just as they do in the heavens.

2. The solar system consists of the sun and all those bodies which revolve round it. These are the planets, with their satellites and the comets.

3. The fixed stars are those bodies which shine by their own light, and are not subject to motion.

The term fixed stars is not strictly correct. Many of the stars are known to have a proper motion through space; and several even of the double stars, besides revolving round each other, are transferred, without parting company, by a progressive motion common to both, to.

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