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the straight line ba. In this position of the earth, the whole of the northern frigid zone is illuminated, and the whole of the southern one is in the dark, and the parallels of latitude are most unequally divided; the days in northern latitudes having then their greatest lengths, and, consequently, the nights their least lengths. From the summer solstice to the autumnal equinox, the parts about the poles at which there is continued day or night, and the inequality in the lengths of day and night at other places, continually decrease.

From the autumnal equinox to the vernal, the angle pcS is greater than a right angle, the north pole is turned from the sun, and the south pole towards him, and the lengths of the days and nights are reversed; the nights being longer than the days in the northern hemisphere, and shorter in the southern; the greatest difference being at the winter solstice, when the angle pcS is greatest, and the inclination of the axis pp' to the circle of illumination is again equal to the obliquity of the ecliptic. This position of the axis is represented at the place of the earth near D.

As two great circles mutually bisect each other, the circle of illumination must always bisect the equator, and consequently, at places on the equator, the days and nights must be equal throughout the year, each being 12 hours long. And it follows from what has been said above, that from the equator to the polar circles the length of the longest day must vary from 12 hours to 24 hours; and from the polar circles to the poles, it must vary from 24 hours to six months.

181. Seasons. The amount of heat derived from the sun at any place, at different times in the year, depends, principally, on the length of the day, and the height to which he ascends above the horizon. The longer the day and the greater the altitude he acquires, the greater must be the amount of heat received. The change in the earth's distance from the sun must produce some. effect; but on account of the small eccentricity of the earth's orbit, this change in distance is only a small part of the whole distance, and consequently the difference in the heat received, depending on this cause, cannot be great. Indeed, the sun is in perigee, or nearest the earth, about the 1st of January, which, in northern latitudes, is near the time at which we usually have our coldest weather.

The zenith of a place, at the terrestrial equator, is in the plane of the equator. It is, therefore, obvious, from a reference to Fig. 30, that as the earth revolves on its axis, the zenith of a place at the equator must, during one half the year, pass on one side of the sun's centre, and during the other half on the other side; or, in other words, the sun during one half the year must pass the meridian to the south of the zenith, and during the other half to the north of it; its meridian zenith distance not, however, at any time. exceeding 23° 28'. Consequently, at places at the equator or near it, the intensity of the sun's heat during the middle part of the day must always be great; and as the days are at no time less or much less than 12 hours in length, the temperature is great throughout the year. The greatest intensity of the sun's rays at those places is at or near the equinoxes, when the sun passes the meridian at the zenith or very near it.

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It is further obvious, that at all places within the torrid zone, the sun must, in the course of the year, pass the meridian on opposite sides of the zenith; and, at two periods in the year, it must pass at or very near the zenith. Consequently, not only at the equator, but throughout the torrid zone, there are two seasons in the year at which the sun, when on the meridian, is nearly or quite vertical, and the intensity of his heat is very great.

In either of the temperate zones, the sun passes the meridian at all times in the year, on the same side of the zenith. The difference, therefore, between the least meridian zenith distance, which, in the northern zone, occurs at the summer solstice, and its greatest, which occurs at the winter solstice, must be twice 23° 28', or 47° nearly. In consequence of this large difference in the sun's meridian zenith distances, and of the increased length of the days at the period he approaches nearest the zenith, and diminished length at the time his meridian altitude is least, the difference in temperature at these opposite seasons is necessarily great.

In the frigid zones, the sun can never ascend far above the horizon, and consequently the temperature is always low.

It follows from the preceding paragraphs, that places at and near the equator may be regarded as having two summers and two winters in each year, without much difference in the temperature, which is always high, and that this is the case throughout the torrid

zone; the difference, however, between the summers and one of the winters increasing as the distance from the equator increases.

It is usual in the north temperate zone to regard Spring as commencing at the vernal equinox, or 20th of March; Summer, at the summer solstice, or 21st of June; Autumn, at the autumnal equinox, or 22d of September; and Winter, at the winter solstice, or 21st of December.

182. Duration of Twilight. The time of twilight at any place is the time during which the sun, in his diurnal motion, is between the horizon of the place and a parallel to the horizon at the distance of about 18° below it (88). Let EPR, Fig. 31, be the meridian of a place, Z its zenith, HR the western half of its horizon, FG a parallel to HR at the distance of 18° below, EQ the equator, P its pole, and MN the western half of the sun's diurnal path. Then the time during which the sun is descending from A to B, or while the hour angle increases from ZPA to ZPB, is the time of the evening twilight. As the sun's declination changes but little during a day, the morning and evening twilights of the same day must evidently be nearly of the same length.

The angle APB, when converted into time at the rate of 15° to the hour, expresses the duration of twilight. The magnitude of this angle, and consequently the duration of twilight, evidently depend on the latitude of the place, and the declination of the sun. For the same declination, the twilight is longer as the latitude is greater. At places in northern latitudes, the twilight is longest at the time of the summer solstice; and shortest, when the sun has a few degrees of south declination.* At Philadelphia, and other places whose latitudes are about 40° N., the shortest twilight occurs about the 6th of March and 8th of October.

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When the sun's diurnal path is M'N", meeting the meridian above F, it is evident the twilight must continue all night, as the sun does not then descend so low as FG. This must take place when PN", his distance from the elevated pole, is less than PF, or PH + HF, or the latitude of the place + 18°; or, which amounts to the same, when the latitude is greater than PN" 18°. Now, when the sun has his greatest declination, and of the same name

*This is shown in the Appendix, art. 53.

with the elevated pole, PN" is about 6610, and, consequently, PN" 18°481°. Hence, at a place whose latitude is more than

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4810, the twilight continues all night at the time the sun's declina

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tion is greatest and of the same name with the latitude.

PROBLEMS.

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183. To find the latitude of a place.

1st Method. Let M, M', m or m', Fig. 31, be the point in which the sun, or a fixed star, passes the meridian PRN. Then we have the latitude ZQ ZM + QM ZM' QM = Qm Zm Qm' - Zm'. Hence, calling the zenith distance of the body north or south, according as the zenith is north or south of the body, when the declination of the body and its correct meridian zenith distance are of the same name, their sum will be the latitude, which will be of that name; and when they are of different names, their difference will be the latitude of the same name with the greater quantity; observing, however, that when the body passes the meridian below the pole, the supplement of the declination must be used instead of the declination itself. Consequently, when, from the observed meridian altitude of the sun or a star, the correct altitude has been found, by applying the proper corrections, the latitude is thus very easily obtained.

If two stars be selected, one of which passes the meridian to the south of the zenith, and the other to the north, at about the same altitude, and the latitude be obtained by each, the mean of the two results will be nearly free from any small errors depending on want of accuracy in the centering or adjustment of the instrument used in observing the altitudes, or in the table of refractions. For, as such errors would affect the observed altitudes equally, or nearly so, making them both too great or too small by the same quantity, it is obvious, from the expressions for the latitude in the two cases, that the latitude obtained by one star must be as much too great as that obtained by the other is too small.

2d Method. Let S, Fig. 33, be the position of a star out of the meridian, and let SD be an arc of a great circle perpendicular to the meridian. If the altitude of the star be observed and corrected for refraction, and the time at which the altitude is taken be also

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observed, we shall have given in the triangle ZPS, the two sides PS and ZS and the angle ZPS, to find PZ, the complement of the latitude. For PS is known from the declination of the star, ZS is the complement of the correct altitude, and the angle ZPS, the star's distance from the meridian, is the difference between the star's right ascension and the sidereal time of observation, expressed in degrees. If the observed time is mean solar time, the corresponding sidereal time must be obtained* (142).

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In the right angled triangle PDS we have (App. 49), = cos SPD tang PS.

tang PD

And, from the right angled triangles PDS and ZDS, we have (App. 45),

cos PS cos PD

cos ZS

cos ZD'

cos SD

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cos PS: cos PD :: cos ZS : cos ZD.

or,

The difference between PD and ZD, or their sum, when D falls between P and Z, gives PZ, the complement of the latitude.

It is best to make the observations when the star is near the meridian, as a slight inaccuracy in the observed time does not then sensibly affect the computed latitude. This is not, however, material when it is the Pole star that is observed, as its motion in altitude is, at all times, slow. The star selected should not be one that passes the meridian so near the zenith as to leave a doubt with regard to the side of it on which the perpendicular SD would fall.

There are various other methods of finding the latitude of a place; one of which has been given in a previous article (58).

183.a Given the latitude of a place and the sun's declination, to find the time of his rising or setting.

Let HWR, Fig. 31, be the western half of the horizon, Z the zenith, EQ the equator, and P the elevated pole. Also let NM be parallel to EQ, at a distance equal to the given declination. Then will A, its intersection with HR, be the point of the horizon at which the sun sets, and the hour angle APM, converted into

* In the Nautical Almanac, the sidereal time at mean noon, at Greenwich, is given for each day in the year; and the method of finding it, for any time, at any meridian, is given also.

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