Chapter I: The Earth

Form and Size.

55. _Form of the Earth._--In ordinary language the term _horizon_ denotes the line that bounds the portion of the earth's surface that is visible at any point.

(1) It is well known that the horizon of a plain presents the form of a circle surrounding the observer. If the latter moves, the circle moves also; but its form remains the same, and is modified only when mountains or other obstacles limit the view. Out at sea, the circular form of the horizon is still more decided, and changes only near the coasts, the outline of which breaks the regularity.

Here, then, we obtain a first notion of the rotundity of the earth, since a sphere is the only body which is presented always to us under the form of a circle, from whatever point on its surface it is viewed.

(2) Moreover, it cannot be maintained that the horizon is the vanishing point of distinct vision, and that it is this which causes the appearance of a circular boundary, because the horizon is enlarged when we mount above the surface of the plain. This will be evident from Fig. 65, in which a mountain is depicted in the middle of a plain, whose uniform curvature is that of a sphere. From the foot of the mountain the spectator will have but a very limited horizon. Let him ascend half way, his visual radius extends, is inclined below the first horizon, and reveals a more extended circular area. At the summit of the mountain the horizon still increases; and, if the atmosphere is pure, the spectator will see numerous objects where from the lower stations the sky alone was visible.

This extension of the horizon would be inexplicable if the earth had the form of an extended plane.

(3) The curvature of the surface of the sea manifests itself in a still more striking manner. If we are on the coast at the summit of a hill, and a vessel appears on the horizon (Fig. 66), we see only the tops of the masts and the highest sails; the lower sails and the hull are invisible. As the vessel approaches, its lower part comes into view above the horizon, and soon it appears entire.

In the same manner the sailors from the ship see the different parts of objects on the land appear successively, beginning with the highest. The reason of this will be evident from Fig. 67, where the course of a vessel, seen in profile, is figured on the convex surface of the sea.

As the curvature of the ocean is the same in every direction, it follows that the surface of the ocean is _spherical_. The same is true of the surface of the land, allowance being made for the various inequalities of the surface. From these and various other indications, we conclude that _the earth is a sphere_.

56. _Size of the Earth._--The size of the earth is ascertained by measuring the length of a degree of a meridian, and multiplying this by three hundred and sixty. This gives the circumference of the earth as about twenty-five thousand miles, and its diameter as about eight thousand miles. We know that the two stations between which we measure are one degree apart when the elevation of the pole at one station is one degree greater than at the other.

57. _The Earth Flattened at the Poles._--Degrees on the meridian have been measured in various parts of the earth, and it has been found that they invariably increase in length as we proceed from the equator towards the pole: hence the earth must curve less and less rapidly as we approach the poles; for the less the curvature of a circle, the larger the degrees on it.

58. _The Earth in Space._--In Fig. 68 we have a view of the earth suspended in space. The side of the earth turned towards the sun is illumined, and the other side is in darkness. As the planet rotates on its axis, successive portions of it will be turned towards the sun. As viewed from a point in space between it and the sun, it will present light and dark portions, which will assume different forms according to the portion which is illumined. These different appearances are shown in Fig. 69.

Day and Night.

59. _Day and Night._--The succession of day and night is due to _the rotation of the earth on its axis_, by which a place on the surface of the earth is carried alternately into the sunshine and out of it. As the sun moves around the heavens on the ecliptic, it will be on the celestial equator when at the equinoxes, and 23-1/2° north of the equator when at the summer solstice, and 23-1/2° south of the equator when at the winter solstice.

60. _Day and Night when the Sun is at the Equinoxes._--When the sun is at either equinox, the diurnal circle described by the sun will coincide with the celestial equator; and therefore half of this diurnal circle will be above the horizon at every point on the surface of the globe. At these times _day and night will be equal in every part of the earth_.

The equality of days and nights when the sun is on the celestial
equator is also evident from the following considerations: one-half
of the earth is in sunshine all of the time; when the sun is on the
celestial equator, it is directly over the equator of the earth, and
the illumination extends from pole to pole, as is evident from Figs.
70 and 71, in the former of which the sun is represented as on the
eastern horizon at a place along the central line of the figure, and
in the latter as on the meridian along the same line. In each
diagram it is seen that the illumination extends from pole to pole:
hence, as the earth rotates on its axis, every place on the surface
will be in the sunshine and out of it just half of the time.

61. _Day and Night when the Sun is at the Summer Solstice._--When the sun is at the summer solstice, it will be 23-1/2° north of the celestial equator. The diurnal circle described by the sun will then be 23-1/2° north of the celestial equator; and more than half of this diurnal circle will be above the horizon at all places north of the equator, and less than half of it at places south of the equator: hence _the days will be longer than the nights at places north of the equator, and shorter than the nights at places south of the equator_. At places within 23-1/2° of the north pole, the entire diurnal circle described by the sun will be above the horizon, so that the sun will not set. At places within 23-1/2° of the south pole of the earth, the entire diurnal circle will be below the horizon, so that the sun will not rise.

The illumination of the earth at this time is shown in Figs. 72 and
73. In Fig. 72 the sun is represented as on the western horizon
along the middle line of the figure, and in Fig. 73 as on the
meridian. It is seen at once that the illumination extends 23-1/2°
beyond the north pole, and falls 23-1/2° short of the south pole. As
the earth rotates on its axis, places near the north pole will be in
the sunshine all the time, while places near the south pole will be
out of the sunshine all the time. All places north of the equator
will be in the sunshine longer than they are out of it, while all
places south of the equator will be out of the sunshine longer than
they are in it.

62. _Day and Night when the Sun is at the Winter Solstice._--When the sun is at the winter solstice, it is 23-1/2° south of the celestial equator. The diurnal circle described by the sun is then 23-1/2° south of the celestial equator. More than half of this diurnal circle will therefore be above the horizon at all places south of the equator, and less than half of it at all places north of the equator: hence _the days will be longer than the nights south of the equator, and shorter than the nights at places north of the equator_. At places within 23-1/2° of the south pole, the diurnal circle described by the sun will be entirely above the horizon, and the sun will therefore not set. At places within 23-1/2° of the north pole, the diurnal circle described by the sun will be wholly below the horizon, and therefore the sun will not rise.

The illumination of the earth at this time is shown in Figs. 74 and
75, and is seen to be the reverse of that shown in Figs. 72 and 73.

63. _Variation in the Length of Day and Night._--As long as the sun is north of the equinoctial, the nights will be longer than the days south of the equator, and shorter than the days north of the equator. It is just the reverse when the sun is south of the equator.

The farther the sun is from the equator, the greater is the inequality of the days and nights.

The farther the place is from the equator, the greater the inequality of its days and nights.

When the distance of a place from the _north_ pole is less than the distance of the sun north of the equinoctial, it will have _continuous day without night_, since the whole of the sun's diurnal circle will be above the horizon. A place within the same distance of the _south_ pole will have _continuous night_.

When the distance of a place from the _north_ pole is less than the distance of the sun south of the equinoctial, it will have _continuous night_, since the whole of the sun's diurnal circle will then be below the horizon. A place within the same distance of the _south_ pole will then have _continuous day_.

At the _equator_ the _days and nights are always equal_; since, no matter where the sun is in the heavens, half of all the diurnal circles described by it will be above the horizon, and half of them below it.

64. _The Zones._--It will be seen, from what has been stated above, that the sun will at some time during the year be directly overhead at every place within 23-1/2° of the equator on either side. This belt of the earth is called the _torrid zone_. The torrid zone is bounded by circles called the _tropics_; that of _Cancer_ on the north, and that of _Capricorn_ on the south.

It will also be seen, that, at every place within 23-1/2° of either pole, there will be, some time during the year, a day during which the sun will not rise, or on which it will not set. These two belts of the earth's surface are called the _frigid zones_. These zones are bounded by the _arctic_ circles. The nearer a place is to the poles, the greater the number of days on which the sun does not rise or set.

Between the frigid zones and the torrid zones, there are two belts on the earth which are called the _temperate zones_. The sun is never overhead at any place in these two zones, but it rises and sets every day at every place within their limits.

65. _The Width of the Zones._--The distance the frigid zones extend from the poles, and the torrid zones from the equator, is exactly equal to _the obliquity of the ecliptic_, or the deviation of the axis of the earth from the perpendicular to the plane of its orbit. Were this deviation forty-five degrees, the obliquity of the ecliptic would be forty-five degrees, the torrid zone would extend forty-five degrees from the equator, and the frigid zones forty-five degrees from the poles. In this case there would be no temperate zones. Were this deviation fifty degrees, the torrid and frigid zones would overlap ten degrees, and there would be two belts of ten degrees on the earth, which would experience alternately during the year a torrid and a frigid climate.

Were the axis of the earth perpendicular to the plane of the earth's orbit, there would be no zones on the earth, and no variation in the length of day and night.

66. _Twilight._--Were it not for the atmosphere, the darkness of midnight would begin the moment the sun sank below the horizon, and would continue till he rose again above the horizon in the east, when the darkness of the night would be suddenly succeeded by the full light of day. The gradual transition from the light of day to the darkness of the night, and from the darkness of the night to the light of day, is called _twilight_, and is due to the _diffusion of light from the upper layers of the atmosphere_ after the sun has ceased to shine on the lower layers at night, or before it has begun to shine on them in the morning.

Let _ABCD_ (Fig. 76) represent a portion of the earth, _A_ a point on its surface where the sun _S_ is setting; and let _SAH_ be a ray of light just grazing the earth at _A_, and leaving the atmosphere at the point _H_. The point _A_ is illuminated by the whole reflective atmosphere _HGFE_. The point _B_, to which the sun has set, receives no direct solar light, nor any reflected from that part of the atmosphere which is below _ALH_; but it receives a twilight from the portion _HLF_, which lies above the visible horizon _BF_. The point _C_ receives a twilight only from the small portion of the atmosphere; while at _D_ the twilight has ceased altogether.

67. _Duration of Twilight._--The astronomical limit of twilight is generally understood to be the instant when stars of the sixth magnitude begin to be visible in the zenith at evening, or disappear in the morning.

Twilight is usually reckoned to last until the sun's depression
below the horizon amounts to eighteen degrees: this, however,
varies; in the tropics a depression of sixteen or seventeen degrees
being sufficient to put an end to the phenomenon, while in England a
depression of seventeen to twenty-one degrees is required. The
duration of twilight differs in different latitudes; it varies also
in the same latitude at different seasons of the year, and depends,
in some measure, on the meteorological condition of the atmosphere.
When the sky is of a pale color, indicating the presence of an
unusual amount of condensed vapor, twilight is of longer duration.
This happens habitually in the polar regions. On the contrary,
within the tropics, where the air is pure and dry, twilight
sometimes lasts only fifteen minutes. Strictly speaking, in the
latitude of Greenwich there is no true night from May 22 to July 21,
but constant twilight from sunset to sunrise. Twilight reaches its
minimum three weeks before the vernal equinox, and three weeks after
the autumnal equinox, when its duration is an hour and fifty
minutes. At midwinter it is longer by about seventeen minutes; but
the augmentation is frequently not perceptible, owing to the greater
prevalence of clouds and haze at that season of the year, which
intercept the light, and hinder it from reaching the earth. The
duration is least at the equator (an hour and twelve minutes), and
increases as we approach the poles; for at the former there are two
twilights every twenty-four hours, but at the latter only two in a
year, each lasting about fifty days. At the north pole the sun is
below the horizon for six months, but from Jan. 29 to the vernal
equinox, and from the autumnal equinox to Nov. 12, the sun is less
than eighteen degrees below the horizon; so that there is twilight
during the whole of these intervals, and thus the length of the
actual night is reduced to two months and a half. The length of the
day in these regions is about six months, during the whole of which
time the sun is constantly above the horizon. The general rule is,
_that to the inhabitants of an oblique sphere the twilight is longer
in proportion as the place is nearer the elevated pole, and the sun
is farther from the equator on the side of the elevated pole_.

The Seasons.

68. _The Seasons._--While the sun is north of the celestial equator, places north of the equator are receiving heat from the sun by day longer than they are losing it by radiation at night, while places south of the equator are losing heat by radiation at night longer than they are receiving it from the sun by day. When, therefore, the sun passes north of the equator, the temperature begins to rise at places north of the equator, and to fall at places south of it. The rise of temperature is most rapid north of the equator when the sun is at the summer solstice; but, for some time after this, the earth continues to receive more heat by day than it loses by night, and therefore the temperature continues to rise. For this reason, the heat is more excessive after the sun passes the summer solstice than before it reaches it.

69. _The Duration of the Seasons._--Summer is counted as beginning in June, when the sun is at the summer solstice, and as continuing until the sun reaches the autumnal equinox, in September. Autumn then begins, and continues until the sun is at the winter solstice, in December. Winter follows, continuing until the sun comes to the vernal equinox, in March, when spring begins, and continues to the summer solstice. In popular reckoning the seasons begin with the first day of June, September, December, and March.

The reason why winter is counted as occurring after the winter solstice is similar to the reason why the summer is placed after the summer solstice. The earth north of the equator is losing heat most rapidly at the time of the winter solstice; but for some time after this it loses more heat by night than it receives by day: hence for some time the temperature continues to fall, and the cold is more intense after the winter solstice than before it.

Of course, when it is summer in the northern hemisphere, it is winter in the southern hemisphere, and the reverse. Fig. 77 shows the portion of the earth's orbit included in each season. It will be seen that the earth is at perihelion in the winter season for places north of the equator, and at aphelion in the summer season. This tends to mitigate somewhat the extreme temperatures of our winters and summers.

70. _The Illumination of the Earth at the different Seasons._--Fig. 78 shows the earth as it would appear to an observer at the sun during each of the four seasons; that is to say, the portion of the earth that is receiving the sun's rays. Figs. 79, 80, 81, and 82 are enlarged views of the earth, as seen from the sun at the time of the summer solstice, of the autumnal equinox, of the winter solstice, and of the vernal equinox.

Fig. 83 is, so to speak, a side view of the earth, showing the limit of sunshine on the earth when the sun is at the summer solstice; and Fig. 84, showing the limit of sunshine when the sun is at the autumnal equinox.

71. _Cause of the Change of Seasons._--Variety in the length of day and night, and diversity in the seasons, depend upon _the obliquity of the ecliptic_. Were there no obliquity of the ecliptic, there would be no inequality in the length of day and night, and but slight diversity of seasons. The greater the obliquity of the ecliptic, the greater would be the variation in the length of the days and nights, and the more extreme the changes of the seasons.

Tides.

72. _Tides._--The alternate rise and fall of the surface of the sea twice in the course of a lunar day, or of twenty-four hours and fifty-one minutes, is known as the _tides_. When the water is rising, it is said to be _flood_ tide; and when it is falling, _ebb_ tide. When the water is at its greatest height, it is said to be _high_ water; and when at its least height, _low_ water.

73. _Cause of the Tides._--It has been known to seafaring nations
from a remote antiquity that there is a singular connection between
the ebb and flow of the tides and the diurnal motion of the moon.

This tidal movement in seeming obedience to the moon was a mystery
until the study of the law of gravitation showed it to be due to
_the attraction of the moon on the waters of the ocean_. The reason
why there are two tides a day will appear from Fig. 85. Let _M_ be
the moon, _E_ the earth, and _EM_ the line joining their centres.
Now, strictly speaking, the moon does not revolve around the earth
any more than the earth around the moon; but the centre of each body
moves around the common centre of gravity of the two bodies. The
earth being eighty times as heavy as the moon, this centre is
situated within the former, about three-quarters of the way from its
centre to its surface, at the point _G_. The body of the earth
itself being solid, every part of it, in consequence of the moon's
attraction, may be considered as describing a circle once in a
month, with a radius equal to _EG_. The centrifugal force caused by
this rotation is just balanced by the mean attraction of the moon
upon the earth. If this attraction were the same on every part of
the earth, there would be everywhere an exact balance between it and
the centrifugal force. But as we pass from _E_ to _D_ the attraction
of the moon diminishes, owing to the increased distance: hence at
_D_ the centrifugal force predominates, and the water therefore
tends to move away from the centre _E_. As we pass from _E_ towards
_C_, the attraction of the moon increases, and therefore exceeds the
centrifugal force: consequently at _C_ there is a tendency to draw
the water towards the moon, but still away from the centre _E_. At
_A_ and _B_ the attraction of the moon increases the gravity of the
water, owing to the convergence of the lines _BM_ and _AM_, along
which it acts: hence the action of the moon tends to make the waters
rise at _D_ and _C_, and to fall at _A_ and _B_, causing two tides
to each apparent diurnal revolution of the moon.

74. _The Lagging of the Tides._--If the waters everywhere yielded
immediately to the attractive force of the moon, it would always be
high water when the moon was on the meridian, low water when she was
rising or setting, and high water again when she was on the meridian
below the horizon. But, owing to the inertia of the water, some time
is necessary for so slight a force to set it in motion; and, once in
motion, it continues so after the force has ceased, and until it has
acted some time in the opposite direction. Therefore, if the motion
of the water were unimpeded, it would not be high water until some
hours after the moon had passed the meridian. The free motion of the
water is also impeded by the islands and continents. These deflect
the tidal wave from its course in such a way that it may, in some
cases, be many hours, or even a whole day, behind its time.
Sometimes two waves meet each other, and raise a very high tide. In
some places the tides run up a long bay, where the motion of a large
mass of water will cause an enormous tide to be raised. In the Bay
of Fundy both of these causes are combined. A tidal wave coming up
the Atlantic coast meets the ocean wave from the east, and, entering
the bay with their combined force, they raise the water at the head
of it to the height of sixty or seventy feet.

75. _Spring-Tides and Neap-Tides._--The sun produces a tide as well as the moon; but the tide-producing force of the sun is only about four-tenths of that of the moon. At new and full moon the two bodies unite their forces, the ebb and flow become greater than the average, and we have the _spring-tides_. When the moon is in her first or third quarter, the two forces act against each other; the tide-producing force is the difference of the two; the ebb and flow are less than the average; and we have the _neap-tides_.

Fig. 86 shows the tide that would be produced by the moon alone; Fig. 87, the tide produced by the combined action of the sun and moon; and Fig. 88, by the sun and moon acting at right angles to each other.

The tide is affected by the distance of the moon from the earth, being highest near the time when the moon is in perigee, and lowest near the time when she is in apogee. When the moon is in perigee, at or near the time of a new or full moon, unusually high tides occur.

76. _Diurnal Inequality of Tides._--The height of the tide at a
given place is influenced by the declination of the moon. When the
moon has no declination, the highest tides should occur along the
equator, and the heights should diminish from thence toward the
north and south; but the two daily tides at any place should have
the same height. When the moon has north declination, as shown in
Fig. 89, the highest tides on the side of the earth next the moon
will be at places having a corresponding north latitude, as at _B_,
and on the opposite side at those which have an equal south
latitude. Of the two daily tides at any place, that which occurs
when the moon is nearest the zenith should be the greatest: hence,
when the moon's declination is north, the height of the tide at a
place in north latitude should be greater when the moon is above the
horizon than when she is below it. At the same time, places south of
the equator have the highest tides when the moon is below the
horizon, and the least when she is above it. This is called the
_diurnal inequality_, because its cycle is one day; but it varies
greatly in amount at different places.

77. _Height of Tides._--At small islands in mid-ocean the tides never rise to a great height, sometimes even less than one foot; and the average height of the tides for the islands of the Atlantic and Pacific Oceans is only three feet and a half. Upon approaching an extensive coast where the water is shallow, the height of the tide is increased; so that, while in mid-ocean the average height does not exceed three feet and a half, the average in the neighborhood of continents is not less than four or five feet.

The Day and Time.

78. _The Day._--By the term _day_ we sometimes denote the period of sunshine as contrasted with that of the absence of sunshine, which we call _night_, and sometimes the period of the earth's rotation on its axis. It is with the latter signification that the term is used in this section. As the earth rotates on its axis, it carries the meridian of a place with it; so that, during each complete rotation of the earth, the portion of the meridian which passes overhead from pole to pole sweeps past every star in the heavens from west to east. The _interval between two successive passages of this portion of the meridian across the same star_ is the exact period of the complete rotation of the earth. This period is called a _sidereal day_. The sidereal day may also be defined as _the interval between two successive passages of the same star across the meridian_; the passage of the meridian across the star, and the passage or _transit_ of the star across the meridian, being the same thing looked at from a different point of view. The interval _between two successive passages of the meridian across the sun_, or _of the sun across the meridian_, is called a _solar day_.

79. _Length of the Solar Day._--The solar day is a little longer than 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.

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.

(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.

(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.

(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.

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.

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.

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.