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To all places in the frigid zones the sun, in summer, does not set for a certain number of days; nor rise in winter for the same number of days: at other times of the year it rises and sets daily.
To all places in the torrid zone the sun is vertical at noon twice in the year: thus an inhabitant of the equator has the sun vertical when it is in the equinoctial. And, at any other period, the places to which the sun is vertical are those whose latitude is equal to the declination of the sun, and of the same name with it; thus, at 10° N. latitude the sun is vertical when its declination is 10° N.
This may be further illustrated by observing, that the equator and equinoctial coinciding (i. e., the equinoctial being nothing more than the equator supposed to be continued to the heavens) when the sun is in the equinoctial, a perpendicular ray, coming from it to the earth, will fall upon the equator: and during a diurnal revolution of the earth, the equator will be formed or passed over by this ray. When the sun is not in the equinoctial, the perpendicular ray will fall as far to the north or south of the equator as the sun is distant north or south of the equinoctial; and during a diurnal revolution of the earth, that parallel of latitude will be described by this ray whose distance from the equator is equal to the sun's declination, and of the same name with it.
Whilst the earth, in its annual motion round the sun, is moving from cancer to capricorn, the sun appears to move from capricorn to cancer; hence its declination varies from 231 S. to 233 N.: and during that time, or in half a year, its rays will have been successively perpendicular to all places in the torrid zone.
Whilst the earth is moving through the other half of its orbit from capricorn to cancer, the sun appears to move from cancer to capricorn, and varies in declination from 231 N, to 23] S.; it will be vertical to the same places, but in a retrograde order.
The tropic of cancer is the most northern circle described by the vertical rays of the sun; that of capricorn the most southern. The sun is vertical only once a year at the tropics; at the tropic of cancer on June 21st, and at the tropic of capricorn on December 21st.
All places out of the torrid zone, being at a greater distance from the equator than the sun's greatest declination, can never have the sur vertical.
From the ecliptic being drawn upon the terrestrial globe, and the pupil knowing that it is the line in which the sun appears to more, he may be inclined to suppose that the sun moves daily round the earth in the oblique manner in which the ecliptic is drawn.
To correct this false notion, it may be observed, that the ecliptic is a circle peculiar to the celestial globe, where it really inarks out the sun's apparent path among the stars; but on the terrestrial globe it is of no other use than to find the sun's declination on any day.
The sun's vertical rays form a sort of spiral line from tropic to tropic. This may be explained by supposing a quantity of silk string to be wrapped round the globe, from one tropic to the other. If the silk string be so contrived as to be thicker towards the equator, where the daily difference of declination is greatest; and if the number of times it requires to be wrapped round, before it covers the space between the tropics, be equal to half the number of days in one year, it will exactly represent the spiral line formed by the rays of the sun in six months. In the other six months the same sort of spiral line will be formed in a contrary direction.
PROBLEM XXII. A place being given in the Torrid Zone, to find those two days of the year in which the Sun is vertical to that place.
By the GLOBE.-1. Bring the given place to the meridian, and mark its latitude.
2. Turn the globe round, and observe the two points of the ecliptic that pass under this mark; the calendar will shew the days corresponding to these points,—which will be the answer required.
Or, having found the latitude of the place, bring the analemma to the meridian, then directly below the latitude will be found the days required.
WITHOUT THE GLOBE.— 1. Find the latitude, either from a table of latitudes and longitudes, or from maps.
2. Observe in White's Ephemeris, or in Table I. at the
end of this work, on what two days of the year the sun's declination is equal to the latitude, and of the same name with it: these are the days required.
The examples to this problem may be proved by the following me.. thod. Find how many days there are from the time when the sun is vertical to the nearest solstice, and also how many there are from that solstice to the time when it is vertical again : if the number of days be equal, the solution is right.
1. Otaheite? Answ. Jan. 30 and Nov. 11.
14. Friendly Isles. 5. Bencoolen.
15. Trincomalee. 6. Quito.
16. Guadaloupe. 7. Borneo.
17. Porto Bello. 8. Pelew Islands.
18. Vera Cruz. 9. Sierra Leone.
19. Tinian Isle. 10. Tobago.
20. Manilla. 11. Port Royal.
21. Columbo. 12. Bombay.
22. Santa Fé de Bogotà
PROBLEM XXIII. To find all those places in the Torrid Zone to which the Sun
is vertical on a given day. Find the sun's declination for the given day, and mark the declination, then turn the globe round, and all those places which pass under that mark of the meridian will have the sun vertical on the given day.
EXAMPLES. 1. To what places is the sun vertical Nov. 10th ?
Answ. To Otaheite, the Great Cyclades, and New Hebrides, in the South Sea; Cape Grafton, in New South Wales; the Island of Madagascar; Monomotapa and
Mataman, in Africa; Punta Gorda, in Brazil; and the southern parts of Amazonia and Peru, in South America.
2. To what places is the sun vertical on February 2nd ? Answ. To the same as in the last example.
To what places is the sun vertical3. On April 16 and Aug. 28 ? 6. May 16 and July 29 ? 4. At the summer solstice? 7. Winter solstice? 5. March 21 and Sept. 23 ? 8. May 1?
PROBLEM XXIV. The day and hour at any place being given, to find where
the Sun is then vertical. Find the sun's declination, and by Prob. XIII. the places where it is noon at the time; then of those places where it is noon, that will have the sun vertical, whose latitude is the same as the declination.
1. To what place is the sun vertical, when it is 39 min. past 6 a.m. at London, August 18th ? Answ. Madras.
2. Where is the sun vertical on the 24th of October, when it is 29 min. past 7 p.m. at Jerusalem ? Ans. Lima.
Having the times given at the following places, where is the sun then vertical ?
3. September 23, 6 50 a.m. Bagdad.
0 30 p.m. Canton.
4, 8 10 a.m. Vienna. 10. May 20, 11 43 p.m. Calicut. 11. January 1, 6 Op.m. Mexico. 12. February 12, 90 a.m. Dublin, 13. July 28, 516 p.m. Port Royal. 14. March 11, 6 10 a.m. Malta.
PROBLEM XXV. Having the hour given at any place, on any given day, to
find where the Sun is rising, where it is setting, where it is noon, and where it is midnight.
Find, by Problem XXIV., the place to which the sun is then vertical ; elevate the globe for that place, and bring it to the meridian.
Then, to all those places in the western semicircle of the horizon, the sun is rising; to those under the upper semicircle of the meridian it is noon; to all places in the eastern semicircle of the horizon the sun is setting ; and to those under the lower semicircle of the meridian it is midnight.
1. At what places is the sun rising on June 4, when it is 4 p.m. at London ? Ans. At the north-east part of Siberia; at Kamtchatka; at the most westerly of the Sandwich Isles; and the most easterly of the Society Isles.
Where is it noon at the same time?
Ans. At Baffin's Bay, New Britain, Martinique, Trinidad, and the middle part of South America.
At what places is the sun then setting ?
Ans. At Tobolsk, the Caspian Sea, Desert of Arabia, the middle of the Red Sea, Abyssinia, the unknown parts of Africa, and the country of the Hottentots.
At what places is it midnight at that time?
Ans. At Chinese Tartary, the eastern part of China, the Philippine Isles, and the western part of New Holland.
2. On April 27th, 6 hrs. 45 min. a.m. at Newcastle,