the sidereal day. This is owing to the sun's eastward motion among the
stars. We have already seen that the sun's apparent position among the
stars is continually shifting towards the east at a rate which causes it
to make a complete circuit of the heavens in a year, or three hundred
and sixty-five days. This is at the rate of about one degree a day:
hence, were the sun and a star on the meridian together to-day, when the
meridian again came around to the star, the sun would appear about one
degree to the eastward: hence the meridian must be carried about one
degree farther in order to come up to the sun. The solar day must
therefore be _about four minutes longer_ than the sidereal day.
[Illustration: Fig. 90.]
[Illustration: Fig. 91.]
The fact that the earth must make more than a complete rotation is also
evident from Figs. 90 and 91. In Fig. 90, _ba_ represents the plane of
the meridian, and the small arrows indicate the direction the earth is
rotating on its axis, and revolving in its orbit. When the earth is at
1, the sun is on the meridian at _a_. When the earth has moved to 2, it
has made a complete rotation, as is shown by the fact that the plane of
the meridian is parallel with its position at 1; but it is evident that
the meridian has not yet come up with the sun. In Fig. 91, _OA_
represents the plane of the meridian, and _OS_ the direction of the sun.
The small arrows indicate the direction of the rotation and revolution
of the earth. In passing from the first position to the second the earth
makes a complete rotation, but the meridian is not brought up to the
sun.
80. _Inequality in the Length of Solar Days._--The sidereal days are all
of the same length; but the solar days differ somewhat in length. This
difference is due to the fact that the sun's apparent position moves
eastward, or _away from the meridian_, at a variable rate.
There are three reasons why this rate is variable:--
(1) The sun's eastward motion is due to the revolution of the earth
in its orbit. Now, the earth's orbital motion is _not uniform_,
being fastest when the earth is at perihelion, and slowest when the
earth is at aphelion: hence, other things being equal, solar days
will be longest when the earth is at perihelion, and shortest when
the earth is at aphelion.
[Illustration: Fig. 92.]
[Illustration: Fig. 93.]
(2) The sun's eastward motion is along the ecliptic. Now, from Figs.
92 and 93, it will be seen, that, when the sun is at one of the
equinoxes, it will be moving away from the meridian _obliquely_;
and, from Figs. 94 and 95, that, when the sun is at one of the
solstices, it will be moving away from the meridian
_perpendicularly_: hence, other things being equal, the sun would
move away from the meridian _fastest_, and the days be _longest_,
when the sun is at the _solstices_; while it would move away from
the meridian _slowest_, and the days be _shortest_, when the sun is
at the _equinoxes_. That a body moving along the ecliptic must be
moving at a variable angle to the meridian becomes very evident on
turning a celestial globe so as to bring each portion of the
ecliptic under the meridian in turn.
[Illustration: Fig. 94.]
[Illustration: Fig. 95.]
(3) The sun, moving along the ecliptic, always moves _in a great
circle_, while the point of the meridian which is to overtake the
sun moves in a diurnal circle, which is _sometimes a great circle_
and _sometimes a small circle_. When the sun is at the equinoxes,
the point of the meridian which is to overtake it moves in a great
circle. As the sun passes from the equinoxes to the solstices, the
point of the meridian which is to overtake it moves on a smaller and
smaller circle: hence, as we pass away from the celestial equator,
the points of the meridian move slower and slower. Therefore, other
things being equal, the meridian will gain upon the sun _most
rapidly_, and the days be _shortest_, when the sun is at the
_equinoxes_; while it will gain on the sun _least rapidly_, and the
days will be _longest_, when the sun is at the _solstices_.
The ordinary or _civil day_ is the mean of all the solar days in a year.
81. _Sun Time and Clock Time._--It is noon by the sun when the sun is on
the meridian, and by the clock at the middle of the civil day. Now, as
the civil days are all of the same length, while solar days are of
variable length, it seldom happens that the middles of these two days
coincide, or that sun time and clock time agree. The difference between
sun time and clock time, or what is often called _apparent solar time_
and _mean solar time_, is called the _equation of time_. The sun is said
to be _slow_ when it crosses the meridian after noon by the clock, and
_fast_ when it crosses the meridian before noon by the clock. Sun time
and clock time coincide four times a year; during two intermediate
seasons the clock time is ahead, and during two it is behind.
* * * * *
The following are the dates of coincidence and of maximum deviation,
which vary but slightly from year to year:--
February 10 True sun fifteen minutes slow.
April 15 True sun correct.
May 14 True sun four minutes fast.
June 14 True sun correct.
July 25 True sun six minutes slow.
August 31 True sun correct.
November 2 True sun sixteen minutes fast.
December 24 True sun correct.
One of the effects of the equation of time which is frequently
misunderstood is, that the interval from sunrise until noon, as given in
the almanacs, is not the same as that between noon and sunset. The
forenoon could not be longer or shorter than the afternoon, if by "noon"
we meant the passage of the sun across the meridian; but the noon of our
clocks being sometimes fifteen minutes before or after noon by the sun,
the former may be half an hour nearer to sunrise than to sunset, or
_vice versa_.
The Year.
82. _The Year._--The _year_ is the time it takes the earth to revolve
around the sun, or, what amounts to the same thing, _the time it takes
the sun to pass around the ecliptic_.
(1) The time it takes the sun to pass from a star around to the same
star again is called a _sidereal year_. This is, of course, the exact
time it takes the earth to make a complete revolution around the sun.
[Illustration: Fig. 96.]
(2) The time it takes the sun to pass around from the vernal equinox, or
the _first point of Aries_, to the vernal equinox again, is called the
_tropical_ year. This is a little shorter than the sidereal year, owing
to the precession of the equinoxes. This will be evident from Fig. 96.
The circle represents the ecliptic, _S_ the sun, and _E_ the vernal
equinox. The sun moves around the ecliptic _eastward_, as indicated by
the long arrow, while the equinox moves slowly _westward_, as indicated
by the short arrow. The sun will therefore meet the equinox before it
has quite completed the circuit of the heavens. The exact lengths of
these respective years are:--
Sidereal year 365.25636=365 days 6 hours 9 min 9 sec
Tropical year 365.24220=365 days 5 hours 48 min 46 sec
Since the recurrence of the seasons depends on the tropical year, the
latter is the one to be used in forming the calendar and for the
purposes of civil life generally. Its true length is eleven minutes and
fourteen seconds less than three hundred and sixty-five days and a
fourth.
It will be seen that the tropical year is about twenty minutes shorter
than the sidereal year.
(3) The time it takes the earth to pass from its perihelion point
around to the perihelion point again is called the _anomalistic
year_. This year is about four minutes longer than the sidereal
year. This is owing to the fact that the major axis of the earth's
orbit is slowly moving around to the east at the rate of about ten
seconds a year. This causes the perihelion point _P_ (Fig. 97) to
move _eastward_ at that rate, as indicated by the short arrow. The
earth _E_ is also moving eastward, as indicated by the long arrow.
Hence the earth, on starting at the perihelion, has to make a little
more than a complete circuit to reach the perihelion point again.
[Illustration: Fig. 97.]
83. _The Calendar._--The _solar year_, or the interval between two
successive passages of the same equinox by the sun, is 365 days, 5
hours, 48 minutes, 46 seconds. If, then, we reckon only 365 days to
a common or _civil year_, the sun will come to the equinox 5 hours,
48 minutes, 46 seconds, or nearly a quarter of a day, later each
year; so that, if the sun entered Aries on the 20th of March one
year, he would enter it on the 21st four years after, on the 22d
eight years after, and so on. Thus in a comparatively short time the
spring months would come in the winter, and the summer months in the
spring.
Among different ancient nations different methods of computing the
year were in use. Some reckoned it by the revolution of the moon,
some by that of the sun; but none, so far as we know, made proper
allowances for deficiencies and excesses. Twelve moons fell short of
the true year, thirteen exceeded it; 365 days were not enough, 366
were too many. To prevent the confusion resulting from these errors,
Julius Cæsar reformed the calendar by making the year consist of 365
days, 6 hours (which is hence called a _Julian_ year), and made
every fourth year consist of 366 days. This method of reckoning is
called _Old Style_.
But as this made the year somewhat too long, and the error in 1582
amounted to ten days, Pope Gregory XIII., in order to bring the
vernal equinox back to the 21st of March again, ordered ten days to
be struck out of that year, calling the next day after the 4th of
October the 15th; and, to prevent similar confusion in the future,
he decreed that three leap-years should be omitted in the course of
every four hundred years. This way of reckoning time is called _New
Style_. It was immediately adopted by most of the European nations,
but was not accepted by the English until the year 1752. The error
then amounted to eleven days, which were taken from the month of
September by calling the 3d of that month the 14th. The Old Style is
still retained by Russia.
According to the Gregorian calendar, _every year whose number is
divisible by four_ is a _leap-year_, except, that, _in the case of
the years whose numbers are exact hundreds, those only are
leap-years which are divisible by four after cutting off the last
two figures_. Thus the years 1600, 2000, 2400, etc., are leap-years;
1700, 1800, 1900, 2100, 2200, etc., are not. The error will not
amount to a day in over three thousand years.
84. _The Dominical Letter._--The _dominical letter_ for any year is
that which we often see placed against Sunday in the almanacs, and
is always one of the first seven in the alphabet. Since a common
year consists of 365 days, if this number is divided by seven (the
number of days in a week), there will be a remainder of one: hence a
year commonly begins one day later in the week than the preceding
one did. If a year of 365 days begins on Sunday, the next will begin
on Monday; if it begins on Thursday, the next will begin on Friday;
and so on. If Sunday falls on the 1st of January, the _first_ letter
of the alphabet, or _A_, is the _dominical letter_. If Sunday falls
on the 7th of January (as it will the next year, unless the first is
leap-year), the _seventh_ letter, _G_, is the dominical letter. If
Sunday falls on the 6th of January (as it will the third year,
unless the first or second is leap-year), the _sixth_ letter, _F_,
will be the dominical letter. Thus, if there were no leap-years, the
dominical letters would regularly follow a retrograde order, _G_,
_F_, _E_, _D_, _C_, _B_, _A_.
But _leap_-years have 366 days, which, divided by seven, leaves two
remainder: hence the years following leap-years will begin two days
later in the week than the leap-years did. To prevent the
interruption which would hence occur in the order of the dominical
letters, leap-years have _two_ dominical letters, one indicating
Sunday till the 29th of February, and the other for the rest of the
year.
By _Table I._ below, the dominical letter for any year (New Style) for
four thousand years from the beginning of the Christian Era may be
found; and it will be readily seen how the Table could be extended
indefinitely by continuing the centuries at the top in the same order.
To find the dominical letter by this table, _look for the hundreds of
years at the top, and for the years below a hundred, at the left hand_.
Thus the letter for 1882 will be opposite the number 82, and in the
column having 1800 at the top; that is, it will be _A_. In the same way,
the letters for 1884, which is a leap-year, will be found to be _FE_.
Having the dominical letter of any year, _Table II._ shows what days of
every month of the year will be _Sundays_.
To find the Sundays of any month in the year by this table, _look in the
column, under the dominical letter, opposite the name of the month given
at the left_.
From the Sundays the date of any other day of the week can be readily
found.
Thus, if we wish to know on what day of the week Christmas falls in
1889, we look opposite December, under the letter _F_ (which we have
found to be the dominical letter for the year), and find that the 22d of
the month is a Sunday; the 25th, or Christmas, will then be Wednesday.
In the same way we may find the day of the week corresponding to any
date (New Style) in history. For instance, the 17th of June, 1775, the
day of the fight at Bunker Hill, is found to have been a _Saturday_.
These two tables then serve as a _perpetual almanac_.
TABLE I.
100 200 300 400
500 600 700 800
900 1000 1100 1200
1300 1400 1500 1600
1700 1800 1900 2000
2100 2200 2300 2400
--- --- --- ----
C E G BA
1 29 57 85 B D F G
2 30 58 86 A C E F
3 31 59 87 G B D E
4 32 60 88 FE AG CB DC
5 33 61 89 D F A B
6 34 62 90 C E G A
7 35 63 91 B D F G
8 36 64 92 AG CB ED FE
9 37 65 93 F A C D
10 38 66 94 E G B C
11 39 67 95 D F A B
12 40 68 96 CB ED GF AG
13 41 69 97 A C E F
14 42 70 98 G B D E
15 43 71 99 F A C D
16 44 72 .. ED GF BA CB
17 45 73 .. C E G A
18 46 74 .. B D F G
19 47 75 .. A C E F
20 48 76 .. GF BA DC ED
21 49 77 .. E G B C
22 50 78 .. D F A B
23 51 79 .. C E G A
24 52 80 .. BA DC FE GF
25 53 81 .. G B D E
26 54 82 .. F A C D
27 55 83 .. E G B C
28 56 84 .. DC FE AG BA
TABLE II.
A B C D E F G
1 2 3 4 5 6 7
Jan. 31. 8 9 10 11 12 13 14
15 16 17 18 19 20 21
Oct. 31. 22 23 24 25 26 27 28
29 30 31 .. .. .. ..
Feb. 28-29. .. .. .. 1 2 3 4
5 6 7 8 9 10 11
March 31. 12 13 14 15 16 17 18
19 20 21 22 23 24 25
Nov. 30. 26 27 28 29 30 31 ..
.. .. .. .. .. .. 1
April 30. 2 3 4 5 6 7 8
9 10 11 12 13 14 15
July 31 16 17 18 19 20 21 22
23 24 25 26 27 28 29
30 31 .. .. .. .. ..
.. .. 1 2 3 4 5
6 7 8 9 10 11 12
Aug. 31. 13 14 15 16 17 18 19
20 21 22 23 24 25 26
27 28 29 30 31 .. ..
.. .. .. .. .. 1 2
Sept. 30. 3 4 5 6 7 8 9
10 11 12 13 14 15 16
17 18 19 20 21 22 23
Dec. 31. 24 25 26 27 28 29 30
31 .. .. .. .. .. ..
.. 1 2 3 4 5 6
7 8 9 10 11 12 13
May. 31. 14 15 16 17 18 19 20
21 22 23 24 25 26 27
28 29 30 31 .. .. ..
.. .. .. .. 1 2 3
4 5 6 7 8 9 10
June 30. 11 12 13 14 15 16 17
18 19 20 21 22 23 24
25 26 27 28 29 30 ..
Weight of the Earth and Precession.
85. _The Weight of the Earth._--There are several methods of
ascertaining the weight and mass of the earth. The simplest, and perhaps
the most trustworthy method is to compare the pull of the earth upon a
ball of lead with that of a known mass of lead upon it. The pull of a
known mass of lead upon the ball may be measured by means of a torsion
balance. One form of the balance employed for this purpose is shown in
Figs. 98 and 99. Two small balls of lead, _b_ and _b_, are fastened to
the ends of a light rod _e_, which is suspended from the point _F_ by
means of the thread _FE_. Two large balls of lead, _W_ and _W_, are
placed on a turn-table, so that one of them shall be just in front of
one of the small balls, and the other just behind the other small ball.
The pull of the large balls turns the rod around a little so as to bring
the small balls nearer the large ones. The small balls move towards the
large ones till they are stopped by the torsion of the thread, which is
then equal to the pull of the large balls. The deflection of the rod is
carefully measured. The table is then turned into the position indicated
by the dotted lines in Fig. 99, so as to reverse the position of the
large balls with reference to the small ones. The rod is now deflected
in the opposite direction, and the amount of deflection is again
carefully measured. The second measurement is made as a check upon the
accuracy of the first. The force required to twist the thread as much as
it was twisted by the deflection of the rod is ascertained by
measurement. This gives the pull of the two large balls upon the two
small ones. We next calculate what this pull would be were the balls as
far apart as the small balls are from the centre of the earth. We can
then form the following proportion: the pull of the large balls upon the
small ones is to the pull of the earth upon the small ones as the mass
of the large balls is to the mass of the earth, or as the weight of the
large balls is to the weight of the earth. Of course, the pull of the
earth upon the small balls is the weight of the small balls. In this way
it has been ascertained that the mass of the earth is about 5.6 times
that of a globe of water of the same size. In other words, the _mean
density_ of the earth is about 5.6.
[Illustration: Fig. 98.]
[Illustration: Fig. 99.]
The weight of the earth in pounds may be found by multiplying the number
of cubic feet in it by 62-1/2 (the weight, in pounds, of one cubic foot
of water), and this product by 5.6.
[Illustration: Fig. 100.]
86. _Cause of Precession._--We have seen that the earth is flattened at
the poles: in other words, the earth has the form of a sphere, with a
protuberant ring around its equator. This equatorial ring is inclined to
the plane of the ecliptic at an angle of about 23-1/2°. In Fig. 100 this
ring is represented as detached from the enclosed sphere. _S_ represents
the sun, and _Sc_ the ecliptic. As the point _A_ of the ring is nearer
the sun than the point _B_ is, the sun's pull upon _A_ is greater than
upon _B_: hence the sun tends to pull the ring over into the plane of
the ecliptic; but the rotation of the earth tends to keep the ring in
the same plane. The struggle between these two tendencies causes the
earth, to which the ring is attached, to wabble like a spinning-top,
whose rotation tends to keep it erect, while gravity tends to pull it
over. The handle of the top has a gyratory motion, which causes it to
describe a curve. The axis of the heavens corresponds to the handle of
the top.
II. THE MOON.
Distance, Size, and Motions.
87. _The Distance of the Moon._--The moon is the nearest of the heavenly
bodies. Its distance from the centre of the earth is only about sixty
times the radius of the earth, or, in round numbers, two hundred and
forty thousand miles.
The ordinary method of finding the distance of one of the nearer
heavenly bodies is first to ascertain its horizontal parallax. This
enables us to form a right-angled triangle, the lengths of whose
sides are easily computed, and the length of whose hypothenuse is
the distance of the body from the centre of the earth.
[Illustration: Fig. 101.]
Horizontal parallax has already been defined (32) as the
displacement of a heavenly body when on the horizon, caused by its
being seen from the surface, instead of the centre, of the earth.
This displacement is due to the fact that the body is seen in a
different direction from the surface of the earth from that in which
it would be seen from the centre. Horizontal parallax might be
defined as the difference in the directions in which a body on the
horizon would be seen from the surface and from the centre of the
earth. Thus, in Fig. 101, _C_ is the centre of the earth, _A_ a
point on the surface, and _B_ a body on the horizon of _A_. _AB_ is
the direction in which the body would be seen from _A_, and _CB_ the
direction in which it would be seen from _C_. The difference of
these directions, or the angle _ABC_, is the parallax of the body.
The triangle _BAC_ is right-angled at _A_; the side _AC_ is the
radius of the earth, and the hypothenuse is the distance of the body
from the centre of the earth. When the parallax _ABC_ is known, the
length of _CB_ can easily by found by trigonometrical computation.
We have seen (32) that the parallax of a heavenly body grows less
and less as the body passes from the horizon towards the zenith. The
parallax of a body and its altitude are, however, so related, that,
when we know the parallax at any altitude, we can readily compute
the horizontal parallax.
The usual method of finding the parallax of one of the nearer
heavenly bodies is first to find its parallax when on the meridian,
as seen from two places on the earth which differ considerably in
latitude: then to calculate what would be the parallax of the body
as seen from one of these places and the centre of the earth: and
then finally to calculate what would be the parallax were the body
on the horizon.
[Illustration: Fig. 102.]
Thus, we should ascertain the parallax of the body _B_ (Fig. 102) as
seen from _A_ and _D_, or the angle _ABD_. We should then calculate
its parallax as seen from _A_ and _C_, or the angle _ABC_. Finally
we should calculate what its parallax would be were the body on the
horizon, or the angle _AB'C_.
The simplest method of finding the parallax of a body _B_ (Fig. 102)
as seen from the two points _A_ and _D_ is to compare its direction
at each point with that of the same fixed star near the body. The
star is so distant, that it will be seen in the same direction from
both points: hence, if the direction of the body differs from that
of the star 2° as seen from one point, and 2° 6' as seen from the
other point, the two lines _AB_ and _DB_ must differ in direction by
6'; in other words, the angle _ABD_ would be 6'.
The method just described is the usual method of finding the
parallax of the moon.
88. _The Apparent Size of the Moon._--The _apparent size_ of a body is
the visual angle subtended by it; that is, the angle formed by two lines
drawn from the eye to two opposite points on the outline of the body.
The apparent size of a body depends upon both its _magnitude_ and its
_distance_.
The apparent size, or _angular diameter_, of the moon is about
thirty-one minutes. This is ascertained by means of the wire micrometer
already described (19). The instrument is adjusted so that its
longitudinal wire shall pass through the centre of the moon, and its
transverse wires shall be tangent to the limbs of the moon on each side,
at the point where they are cut by the longitudinal wire. The micrometer
screw is then turned till the wires are brought together. The number of
turns of the screw needed to accomplish this will indicate the arc
between the wires, or the angular diameter of the moon.
[Illustration: Fig. 103.]
In order to be certain that the longitudinal wire shall pass through the
centre of the moon, it is best to take the moon when its disc is in the
form of a crescent, and to place the longitudinal wire against the
points, or _cusps_, of the crescent, as shown in Fig. 103.
[Illustration: Fig. 104.]
89. _The Real Size of the Moon._--The real diameter of the moon is a
little over one-fourth of that of the earth, or a little more than two
thousand miles. The comparative sizes of the earth and moon are shown in
Fig. 104.
[Illustration: Fig. 105.]
The distance and apparent size of the moon being known, her real
diameter is found by means of a triangle formed as shown in Fig.
105. _C_ represents the centre of the moon, _CB_ the distance of the
moon from the earth, and _CA_ the radius of the moon. _BAC_ is a
triangle, right-angled at _A_. The angle _ABC_ is half the apparent
diameter of the moon. With the angles _A_ and _B_, and the side _CB_
known, it is easy to find the length of _AC_ by trigonometrical
computation. Twice _AC_ will be the diameter of the moon.
The volume of the moon is about one-fiftieth of that of the earth.
90. _Apparent Size of the Moon on the Horizon and in the Zenith._.--The
moon is nearly four thousand miles farther from the observer when she is
on the horizon than when she is in the zenith. This is evident from Fig.
106. _C_ is the centre of the earth, _M_ the moon on the horizon, _M'_
the moon in the zenith, and _O_ the point of observation. _OM_ is the
distance of the moon when she is on the horizon, and _OM'_ the distance
of the moon from the observer when she is in the zenith. _CM_ is equal
to _CM'_, and _OM_ is about the length of _CM_; but _OM'_ is about four
thousand miles shorter than _CM'_: hence _OM'_ is about four thousand
miles shorter than _OM_.
[Illustration: Fig. 106.]
Notwithstanding the moon is much nearer when at the zenith than at
the horizon, it seems to us much larger at the horizon.