Let the student redraw Fig. 28 on a large scale, so that the points _1_ and _3_ shall be only 18° apart, as seen from the earth's center. He will find that the point _L_ is brought down much closer to the surface of the earth, and measuring the length of the line _2 L_, he should find for the "height of the atmosphere" about one-eightieth part of the radius of the earth--i. e., a little less than 50 miles. This, however, is not the true height of the atmosphere. The air extends far beyond this, but the particles of dust and vapor which are capable of sending sunlight down to the earth seem all to lie below this limit.
The student should not fail to watch the eastern sky after sunset, and see the shadow of the earth rise up and fill it while the twilight arch retreats steadily toward the west.
[Illustration: FIG. 29.--The cause of long and short twilights.]
_Duration of twilight._--Since twilight ends when the sun is 18° below the horizon, any circumstance which makes the sun go down rapidly will shorten the duration of twilight, and anything which retards the downward motion of the sun will correspondingly prolong it. Chief among influences of this kind is the angle which the sun's course makes with the horizon. If it goes straight down, as at _a_, Fig. 29, a much shorter time will suffice to carry it to a depression of 18° than is needed in the case shown at _b_ in the same figure, where the motion is very oblique to the horizon. If we consider different latitudes and different seasons of the year, we shall find every possible variety of circumstance from _a_ to _b_, and corresponding to these, the duration of twilight varies from an all-night duration in the summers of Scotland and more northern lands to an hour or less in the mountains of Peru. For the sake of graphical effect, the shortness of tropical twilight is somewhat exaggerated by Coleridge in the lines,
"The sun's rim dips; the stars rush out: At one stride comes the dark." _The Ancient Mariner._
In the United States the longest twilights come at the end of June, and last for a little more than two hours, while the shortest ones are in March and September, amounting to a little more than an hour and a half; but at all times the last half hour of twilight is hardly to be distinguished from night, so small is the quantity of reflecting matter in the upper regions of the atmosphere. For practical convenience it is customary to assume in the courts of law that twilight ends an hour after sunset.