THE MEASUREMENT OF TIME
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.
[Illustration: 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 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--
_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 [b]
Cassiopeiæ, 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 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.
[Illustration: FIG. 31.--Longitude and time]
[Illustration: 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 passing 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
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 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 gods, 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.
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.--
One tropical year = 365.242197 mean solar days.
" " " = 365d. 5h. 48m. 45.8s.
One lunation = 29.530588 mean solar days.
" " = 29d. 12h. 44m. 2.8s.
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:
"Thirty days hath September,
April, June, and November;
All the rest have thirty-one
Save February,
Which alone hath twenty-eight,
Till leap year gives it twenty-nine."
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 by 4 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
J = Sept. 3
N - 2 = + 15
- N/4 = - 4
------------------
G = Sept. 14
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 by 7.
If _r_ = 1 the date falls on Sunday, etc., and if _r_ = 0 the day is
Saturday. For the example suggested above we have
Jan. 31
Feb. 29
Mch. 31
April 30
May 31
June 30
July 31
Aug. 31
Sept. 30
Oct. 21
---
D = 295
Y = 1492
+ D = + 295
+ (Y - 1) ÷ 4 = + 372
- (Y - 1) ÷ 100 = - 14
+ (Y - 1) ÷ 400 = + 3
-------
7) 2148
_q_ = 306
_r_ = 6 = Friday.
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 coefficient 7? 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.
[Illustration: THE YERKES OBSERVATORY, WILLIAMS BAY, WIS.]