52. Solar time.—To measure any quantity 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 orbital motion around the sun goes faster in some parts of its orbit 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.

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 passed 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—

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 ߠCassiopeiÆ, 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 assumed 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 masses 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 pass, 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 Francisco by 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 page36. 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.

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 passing through the Royal Observatory at Greenwich, England.

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 illustrate 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 orbit. This time of one revolution is called a sidereal year, while, as we have already seen in ChapterV, 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[B] 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:

60. Day, month, and year.—If the day, the month, and the year are to be used concurrently, it is necessary to determine how many days are contained in the month and year, and when this has been done by the astronomer the numbers are found to be very awkward and inconvenient for daily use; and much of the history of chronology consists in an account of the various devices by which ingenious men have sought to use integral numbers to replace the cumbrous decimal fractions which follow.

According to Professor Harkness, for the epoch 1900 A.D.—

The word lunation means the average interval from one new moon to the next one—i.e., the time required by the moon to go from conjunction with the sun round to conjunction again.

A very ancient device was to call a year equal to 365 days, and to have months alternately of 29 and 30 days in length, but this was unsatisfactory in more than one way. At the end of four years this artificial calendar would be about one day ahead of the true one, at the end of forty years ten days in error, and within a single lifetime the seasons would have appreciably changed their position in the year, April weather being due in March, according to the calendar. So, too, the year under this arrangement did not consist of any integral number of months, 12 months of the average length of 29.5 days being 354 days, and 13 months 383.5 days, thus making any particular month change its position from the beginning to the middle and the end of the year within a comparatively short time. Some peoples gave up the astronomical year as an independent unit and adopted a conventional year of 12 lunar months, 354 days, which is now in use in certain Mohammedan countries, where it is known as the wandering year, with reference to the changing positions of the seasons in such a year. Others held to the astronomical year and adopted a system of conventional months, such that twelve of them would just make up a year, as is done to this day in our own calendar, whose months of arbitrary length we are compelled to remember by some such jingle as the following:

61. The calendar.—The foundations of our calendar may fairly be ascribed to Julius CÆsar, who, under the advice of the Egyptian astronomer Sosigines, adopted the old Egyptian device of a leap year, whereby every fourth year was to consist of 366 days, while ordinary years were only 365 days long. He also placed the beginning of the year at the first of January, instead of in March, where it had formerly been, and gave his own name, Julius, to the month which we now call July. August was afterward named in honor of his successor, Augustus. The names of the earlier months of the year are drawn from Roman mythology; those of the later months, September, October, etc., meaning seventh month, eighth month, represent the places of these months in the year, before CÆsar's reformation, and also their places in some of the subsequent calendars, for the widest diversity of practice existed during mediÆval times with regard to the day on which the new year should begin, Christmas, Easter, March 25th, and others having been employed at different times and places.

The system of leap years introduced by CÆsar makes the average length of a year 365.25 days, which differs by about eleven minutes from the true length of the tropical year, a difference so small that for ordinary purposes no better approximation to the true length of the year need be desired. But any deviation from the true length, however small, must in the course of time shift the seasons, the vernal and autumnal equinox, to another part of the year, and the ecclesiastical authorities of mediÆval Europe found here ground for objection to CÆsar's calendar, since the great Church festival of Easter has its date determined with reference to the vernal equinox, and with the lapse of centuries Easter became more and more displaced in the calendar, until Pope Gregory XIII, late in the sixteenth century, decreed another reformation, whereby ten days were dropped from the calendar, the day after March 11th being called March 21st, to bring back the vernal equinox to the date on which it fell in A.D. 325, the time of the Council of NicÆa, which Gregory adopted as the fundamental epoch of his calendar.

The calendar having thus been brought back into agreement with that of old time, Gregory purposed to keep it in such agreement for the future by modifying CÆsar's leap-year rule so that it should run: Every year whose number is divisible by4 shall be a leap year except those years whose numbers are divisible by 100 but not divisible by 400. These latter years—e.g., 1900—are counted as common years. The calendar thus altered is called Gregorian to distinguish it from the older, Julian calendar, and it found speedy acceptance in those civilized countries whose Church adhered to Rome; but the Protestant powers were slow to adopt it, and it was introduced into England and her American colonies by act of Parliament in the year 1752, nearly two centuries after Gregory's time. In Russia the Julian calendar has remained in common use to our own day, but in commercial affairs it is there customary to write the date according to both calendars—e.g., July 4/16, and at the present time strenuous exertions are making in that country for the adoption of the Gregorian calendar to the complete exclusion of the Julian one.

The Julian and Gregorian calendars are frequently represented by the abbreviations O.S. and N.S., old style, new style, and as the older historical dates are usually expressed in O.S., it is sometimes convenient to transform a date from the one calendar to the other. This is readily done by the formula

G = J + (N - 2) - N/4,

where G and J are the respective dates, N is the number of the century, and the remainder is to be neglected in the division by 4. For September 3, 1752, O.S., we have

and September 14 is the date fixed by act of Parliament to correspond to September 3, 1752, O.S. Columbus discovered America on October 12, 1492, O.S. What is the corresponding date in the Gregorian calendar?62. The day of the week.—A problem similar to the above but more complicated consists in finding the day of the week on which any given date of the Gregorian calendar falls—e.g., October 21, 1492.

The formula for this case is

7q + r = Y + D + (Y - 1)/4 - (Y - 1)/100 + (Y - 1)/400

where Y denotes the given year, D the number of the day (date) in that year, and q and r are respectively the quotient and the remainder obtained by dividing the second member of the equation by7. If r=1 the date falls on Sunday, etc., and if r=0 the day is Saturday. For the example suggested above we have

Find from some history the day of the week on which Columbus first saw America, and compare this with the above.

On what day of the week did last Christmas fall? On what day of the week were you born? In the formula for the day of the week why does q have the coefficient7? What principles in the calendar give rise to the divisors 4, 100, 400?

For much curious and interesting information about methods of reckoning the lapse of time the student may consult the articles Calendar and Chronology in any good encyclopÆdia.