Fig. 28 shows a part of the earth surrounded by such a dust-laden atmosphere, which is illuminated on the left by the rays of the sun, but which, on the right of the figure, lies in the shadow cast by the earth.

To an observer placed at _1_ the sun is just setting, and all the atmosphere above him is illumined with its rays, which furnish a bright twilight. When, by the earth's rotation, this observer has been carried to _2_, all the region to the east of his zenith lies in the shadow, while to the west there is a part of the atmosphere from which there still comes a twilight, but now comparatively faint, because the lower part of the atmosphere about our observer lies in the shadow, and it is mainly its upper regions from which the light comes, and here the dust and moisture are much less abundant than in the lower strata. Still later, when the observer has been carried by the earth's rotation to the point _3_, every vestige of twilight will have vanished from his sky, because all of the illuminated part of the atmosphere is now below his horizon, which is represented by the line _3 L_. In the figure the sun is represented to be 78 below this horizon line at the end of twilight, but this is a gross exaggeration, made for the sake of clearness in the drawing--in fact, twilight is usually said to end when the sun is 18 below the horizon.

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.

[Ill.u.s.tration: FIG. 29.--The cause of long and short twilights.]

_Duration of twilight._--Since twilight ends when the sun is 18 below the horizon, any circ.u.mstance which makes the sun go down rapidly will shorten the duration of twilight, and anything which r.e.t.a.r.ds 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 lat.i.tudes and different seasons of the year, we shall find every possible variety of circ.u.mstance 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 quant.i.ty of reflecting matter in the upper regions of the atmosphere. For practical convenience it is customary to a.s.sume in the courts of law that twilight ends an hour after sunset.

How long does twilight last at the north pole?

_The Aurora._--One other phenomenon of the atmosphere may be mentioned, only to point out that it is not of an astronomical character. The Aurora, or northern lights, is as purely an affair of the earth as is a thunderstorm, and its explanation belongs to the subject of terrestrial magnetism.

CHAPTER VI

THE MEASUREMENT OF TIME

52. SOLAR TIME.--To measure any quant.i.ty we need a unit in terms of which it must be expressed. Angles are measured in degrees, and the degree is the unit for angular measurement. For most scientific purposes the centimeter is adopted as the unit with which to measure distances, and similarly a day is the fundamental unit for the measurement of time.

Hours, minutes, and seconds are aliquot parts of this unit convenient for use in dealing with shorter periods than a day, and the week, month, and year which we use in our calendars are multiples of the day.

Strictly speaking, a day is not the time required by the earth to make one revolution upon its axis, but it is best defined as the amount of time required for a particular part of the sky to make the complete circuit from the meridian of a particular place through west and east back to the meridian again. The day begins at the moment when this specified part of the sky is on the meridian, and "the time" at any moment is the hour angle of this particular part of the sky--i. e., the number of hours, minutes, etc., that have elapsed since it was on the meridian.

The student has already become familiar with the kind of day which is based upon the motion of the vernal equinox, and which furnishes sidereal time, and he has seen that sidereal time, while very convenient in dealing with the motions of the stars, is decidedly inconvenient for the ordinary affairs of life since in the reckoning of the hours it takes no account of daylight and darkness. One can not tell off-hand whether 10 hours, sidereal time, falls in the day or in the night. We must in some way obtain a day and a system of time reckoning based upon the apparent diurnal motion of the sun, and we may, if we choose, take the sun itself as the point in the heavens whose transit over the meridian shall mark the beginning and the end of the day. In this system "the time" is the number of hours, minutes, etc., which have elapsed since the sun was on the meridian, and this is the kind of time which is shown by a sun dial, and which was in general use, years ago, before clocks and watches became common. Since the sun moves among the stars about a degree per day, it is easily seen that the rotating earth will have to turn farther in order to carry any particular meridian from the sun around to the sun again, than to carry it from a star around to the same star, or from the vernal equinox around to the vernal equinox again; just as the minute hand of a clock turns farther in going from the hour hand round to the hour hand again than it turns in going from XII to XII. These solar days and hours and minutes are therefore a little longer than the corresponding sidereal ones, and this furnishes the explanation why the stars come to the meridian a little earlier, by solar time, every night than on the night before, and why sidereal time gains steadily upon solar time, this gain amounting to approximately 3m. 56.5s. per day, or exactly one day per year, since the sun makes the complete circuit of the constellations once in a year.

With the general introduction of clocks and watches into use about a century ago this kind of solar time went out of common use, since no well-regulated clock could keep the time correctly. The earth in its...o...b..tal motion around the sun goes faster in some parts of its...o...b..t than in others, and in consequence the sun appears to move more rapidly among the stars in winter than in summer; moreover, on account of the convergence of hour circles as we go away from the equator, the same amount of motion along the ecliptic produces more effect in winter and summer when the sun is north or south, than it does in the spring and autumn when the sun is near the equator, and as a combined result of these causes and other minor ones true solar time, as it is called, is itself not uniform, but falls behind the uniform lapse of sidereal time at a variable rate, sometimes quicker, sometimes slower. A true solar day, from noon to noon, is 51 seconds shorter in September than in December.

[Ill.u.s.tration: FIG. 30.--The equation of time.]

53. MEAN SOLAR TIME.--To remedy these inconveniences there has been invented and brought into common use what is called _mean solar time_, which is perfectly uniform in its lapse and which, by comparison with sidereal time, loses exactly one day per year. "The time" in this system never differs much from true solar time, and the difference between the two for any particular day may be found in any good almanac, or may be read from the curve in Fig. 30, in which the part of the curve above the line marked _0m_ shows how many minutes mean solar time is faster than true solar time. The correct name for this difference between the two kinds of solar time is the _equation of time_, but in the almanacs it is frequently marked "sun fast" or "sun slow." In sidereal time and true solar time the distinction between A. M. hours (_ante meridiem_ = before the sun reaches the meridian) and P. M. hours (_post meridiem_ = after the sun has pa.s.sed the meridian) is not observed, "the time" being counted from 0 hours to 24 hours, commencing when the sun or vernal equinox is on the meridian. Occasionally the attempt is made to introduce into common use this mode of reckoning the hours, beginning the day (date) at midnight and counting the hours consecutively up to 24, when the next date is reached and a new start made. Such a system would simplify railway time tables and similar publications; but the American public is slow to adopt it, although the system has come into practical use in Canada and Spain.

54. TO FIND (APPROXIMATELY) THE SIDEREAL TIME AT ANY MOMENT.--RULE I.

When the mean solar time is known. Let _W_ represent the time shown by an ordinary watch, and represent by _S_ the corresponding sidereal time and by _D_ the number of days that have elapsed from March 23d to the date in question. Then

S = W + 69/70 D 4.

The last term is expressed in minutes, and should be reduced to hours and minutes. Thus at 4 P. M. on July 4th--

_D_ = 103 days.

69/70 _D_ 4 = 406m.

= 6h. 46m.

_W_ = 4h. 0m.

_S_ = 10h. 46m.

The daily gain of sidereal upon mean solar time is 69/70 of 4 minutes, and March 23d is the date on which sidereal and mean solar time are together, taking the average of one year with another, but it varies a little from year to year on account of the extra day introduced in leap years.

RULE II. When the stars in the northern sky can be seen. Find Ca.s.siopeiae, and imagine a line drawn from it to Polaris, and another line from Polaris to the zenith. The sidereal time is equal to the angle between these lines, provided that that angle must be measured from the zenith toward the west. Turn the angle from degrees into hours by dividing by 15.

55. THE EARTH'S ROTATION.--We are familiar with the fact that a watch may run faster at one time than at another, and it is worth while to inquire if the same is not true of our chief timepiece--the earth. It is a.s.sumed in the sections upon the measurement of time that the earth turns about its axis with absolute uniformity, so that mean solar time never gains or loses even the smallest fraction of a second. Whether this be absolutely true or not, no one has ever succeeded in finding convincing proof of a variation large enough to be measured, although it has recently been shown that the axis about which it rotates is not perfectly fixed within the body of the earth. The solid body of the earth wriggles about this axis like a fish upon a hook, so that the position of the north pole upon the earth's surface changes within a year to the extent of 40 or 50 feet (15 meters) without ever getting more than this distance away from its average position. This is probably caused by the periodical shifting of ma.s.ses of air and water from one part of the earth to another as the seasons change, and it seems probable that these changes will produce some small effect upon the rotation of the earth. But in spite of these, for any such moderate interval of time as a year or a century, so far as present knowledge goes, we may regard the earth's rotation as uniform and undisturbed. For longer intervals--e. g., 1,000,000 or 10,000,000 years--the question is a very different one, and we shall have to meet it again in another connection.

56. LONGITUDE AND TIME.--In what precedes there has been constant reference to the meridian. The day begins when the sun is on the meridian. Solar time is the angular distance of the sun past the meridian. Sidereal time was determined by observing transits of stars over a meridian line actually laid out upon the ground, etc. But every place upon the earth has its own meridian from which "the time" may be reckoned, and in Fig. 31, where the rays of sunlight are represented as falling upon a part of the earth's equator through which the meridians of New York, Chicago, and San Francisco pa.s.s, it is evident that these rays make different angles with the meridians, and that the sun is farther from the meridian of New York than from that of San Francis...o...b.. an amount just equal to the angle at _O_ between these meridians. This angle is called by geographers the difference of longitude between the two places, and the student should note that the word longitude is here used in a different sense from that on page 36. From Fig. 31 we obtain the

_Theorem._--The difference between "the times" at any two meridians is equal to their difference of longitude, and the time at the eastern meridian is greater than at the western meridian. Astronomers usually express differences of longitude in hours instead of degrees. 1h. = 15.

The name given to any kind of time should distinguish all the elements which enter into it--e. g., New York sidereal time means the hour angle of the vernal equinox measured from the meridian of New York, Chicago true solar time is the hour angle of the sun reckoned from the meridian of Chicago, etc.

[Ill.u.s.tration: FIG. 31.--Longitude and time]

[Ill.u.s.tration: FIG. 32.--Standard time.]

57. STANDARD TIME.--The requirements of railroad traffic have led to the use throughout the United States and Canada of four "standard times,"

each of which is a mean solar time some integral number of hours slower than the time of the meridian pa.s.sing through the Royal Observatory at Greenwich, England.

Eastern time is 5 hours slower than that of Greenwich.

Central " 6 " " " " "

Mountain " 7 " " " " "

Pacific " 8 " " " " "

In Fig. 32 the broken lines indicate roughly the parts of the United States and Canada in which these several kinds of time are used, and ill.u.s.trate how irregular are the boundaries of these parts.

Standard time is sent daily into all of the more important telegraph offices of the United States, and serves to regulate watches and clocks, to the almost complete exclusion of local time.

58. TO DETERMINE THE LONGITUDE.--With an ordinary watch observe the time of the sun's transit over your local meridian, and correct the observed time for the equation of time by means of the curve in Fig. 30. The difference between the corrected time and 12 o'clock will be the correction of your watch referred to local mean solar time. Compare your watch with the time signals in the nearest telegraph office and find its correction referred to standard time. The difference between the two corrections is the difference between your longitude and that of the standard meridian.

N. B.--Don't tamper with the watch by trying to "set it right." No harm will be done if it is wrong, provided you take due account of the correction as indicated above.

If the correction of the watch changed between your observation and the comparison in the telegraph office, what effect would it have upon the longitude determination? How can you avoid this effect?

59. CHRONOLOGY.--The Century Dictionary defines chronology as "the science of time"--that is, "the method of measuring or computing time by regular divisions or periods according to the revolutions of the sun or moon."

We have already seen that for the measurement of short intervals of time the day and its subdivisions--hours, minutes, seconds--furnish a very complete and convenient system. But for longer periods, extending to hundreds and thousands of days, a larger unit of time is required, and for the most part these longer units have in all ages and among all peoples been based upon astronomical considerations. But to this there is one marked exception. The week is a simple multiple of the day, as the dime is a multiple of the cent, and while it may have had its origin in the changing phases of the moon this is at best doubtful, since it does not follow these with any considerable accuracy. If the still longer units of time--the month and the year--had equally been made to consist of an integral number of days much confusion and misunderstanding might have been avoided, and the annals of ancient times would have presented fewer pitfalls to the historian than is now the case. The month is plainly connected with the motion of the moon among the stars. The year is, of course, based upon the motion of the sun through the heavens and the change of seasons which is thus produced; although, as commonly employed, it is not quite the same as the time required by the earth to make one complete revolution in its...o...b..t. This time of one revolution is called a sidereal year, while, as we have already seen in Chapter V, the year which measures the course of the seasons is shorter than this on account of the precession of the equinoxes. It is called a tropical year with reference to the circuit which the sun makes from one tropic to the other and back again.

We can readily understand why primitive peoples should adopt as units of time these natural periods, but in so doing they incurred much the same kind of difficulty that we should experience in trying to use both English and American money in the ordinary transactions of life. How many dollars make a pound sterling? How shall we make change with English shillings and American dimes, etc.? How much is one unit worth in terms of the other?

One of the Greek poets[2] has left us a quaint account of the confusion which existed in his time with regard to the place of months and moons in the calendar:

"The moon by us to you her greeting sends, But bids us say that she's an ill-used moon And takes it much amiss that you will still Shuffle her days and turn them topsy-turvy, So that when G.o.ds, who know their feast days well, By your false count are sent home supperless, They scold and storm at her for your neglect."

[2] Aristophanes, The Clouds, Whewell's translation.