circle which is above the horizon is the diurnal arc; that which is below, the nocturnal arc. The ratio between them is the ratio between day and night.

133. At the equator, where the equinoctial is perpendicular to the horizon (38), the circles of daily motion. are equally divided; hence the sun is as long above as below the horizon, during each twenty-four hours of the year, and the day is always equal to the night.

134. At the pole, where the equinoctial coincides with the horizon (37), the circles of daily motion are either wholly above, or wholly below the horizon. When the sun's declination is of the same kind as the pole in question, he is above the horizon; when of the opposite kind, he is below. As the sun has north declination half the year, the day at the north pole lasts six months, and the night six months.

135. When the sun is at the equinox, his circle of daily motion is the equinoctial. But the equinoctial and the horizon, being both great circles on the sky (8, 31), divide each other into semicircles (55); hence the day is equal to the night throughout the world, save at the poles; there the sun, being on the horizon, is making the transition between day and night.

136. In north latitude.-The diagram shows the position of the circles of daily motion at 40° north latitude. It is evident that of two circles, parallel to the equinoctial, that which is farthest north has the largest proportional part above the horizon. Hence, at this place:

1. The day is longer than the night when the sun's declination is north, and conversely when it is south.

2. The length of the day increases as the sun moves northward until he reaches his greatest northern declination on the 20th of June (51); the days become shorter

as the sun moves southward until he reaches his greatest southern declination on the 22d of December (52).

137. Farther north.-As the observer goes north from the equator, the angle between the equinoctial and the horizon becomes less; the circles of daily motion lie more obliquely; those north of the equinoctial show a

rapidly increasing diurnal arc; the proportion in those south of the equinoctial decreases as rapidly; long days become longer, and short days shorter. At 66° 32' north latitude the circle of daily motion on the longest day of the year coincides with the circle of perpetual apparition (42); for, as the declination is 23° 28', the

north polar distance is 66° 32′ (34), which is equal to the latitude. On the 20th of June, therefore, the sun does not set. Farther north the sun will not set so long as his declination is more than the distance in degrees and minutes from the place of the observer to the pole, or more than the co-latitude.

138. In the southern hemisphere, all the results described for the northern hemisphere are reversed. The sun passes from east to west through the northern sky; he casts all midday shadows toward the south; the longest days are in December, and the shortest are in June.

139. The amplitude (12) of sunrise and sunset.-The diagram also illustrates the variable position of the sun at sunrise and sunset. When on the equinoctial, the sun rises exactly in the east and sets precisely in the west. As his declination increases northward, the places of both sunrise and sunset move toward the north, and this movement increases with the latitude. At the polar circle, the sun on the longest day merely touches the horizon at the north point, setting and rising again the next instant. On the shortest day, when his declination is south, he appears but for an instant at the south point, rising and setting again immediately.

140. Corrections.-If we seek the exact time or place of sunrise or sunset, corrections must be made :

1. For the effect of atmospheric refraction (129). 2. For the breadth of the sun's disc.

Hitherto reference has always been made to the center of the sun's disc. But sunrise comes at the instant when the first ray crosses the horizon, and sunset is delayed until the last ray vanishes from the upper limb. Hence,

as the sun's disc is 32' broad, sunrise occurs when the sun's center is 16' below the horizon.

The effect of both these corrections is to lengthen the day, and to shorten the night in all parts of the world.

TWILIGHT.

141. Daylight does not instantly vanish at sunset: it fades away gradually, passing through all the shades of waning light which we call twilight. This is caused by the reflection of the light from the upper regions of the air. Let the curve ACEF represent the surface of the atmosphere which

Fig. 48.

surrounds the

earth, and sup

pose the light comes from the sun in the direction of S. No direct rays of sunshine come to the earth beyond atmosphere is illuminated as far as the line BE. The observer at A sees the sun in his horizon. The sun has set for the observer at B, but he sees some reflected sunlight from the space between A and B. For the observer at C both direct and reflected light have vanished.

the line AF, but some portion of the

The twilight which precedes sunrise is called the dawn.

142. Duration of twilight.-Twilight continues until the sun is 18° below the horizon; some writers say 24°. Its duration must vary somewhat with the condition of the air. The zone which is thus partially lighted is about 1250 miles wide.

The duration varies with the latitude. At the equator, where the daily path of the sun is at right angles to the horizon, the twilight zone is passed in 1 h. 12 m. All travelers in equatorial regions remark the very brief time between sunshine and the darkness of night. As the latitude increases, the sun's path crosses the zone more obliquely, and twilight lasts longer. At the pole, twilight continues until the sun has reached a declination 18° on the opposite side of the equinoctial.

143. The crepuscular curve.-Lacaille claimed to have actually seen, while at sea in the South Atlantic, the shadow of the earth forming a curve on the sky opposite the sun, and following the sun toward the west as the twilight faded. This curve which separates the illuminated portion of the sky from the darker part is called the crepuscular curve. If seen at all, it must be under

the most favorable circumstances and in the clearest air.

144. The height of the atmosphere.-Twilight would last longer if the layer of air about the earth were thicker, or if its upper strata were denser than it now is. If the crepuscular curve could be clearly seen, the thickness of the atmosphere might be easily computed; on the supposition that twilight lasts until the sun is 18° below the horizon, the height of the atmosphere would be about 40 miles. But the density of the air diminishes with its height above the earth, and in its upper regions it doubtless becomes too rare to reflect much light, if any. Hence the air may be presumed to extend considerably above that distance.

Without the quality in the air which produces the diffusion or dispersion of light, there could be no twilight; every place not in direct sunshine would be utterly dark, even in the daytime.