horizon towards the zenith. This is evident from Fig. 42. _OD_ is the
horizon, and _Z_ the zenith, of an observer at _O_. Refraction would
make a star at _Q_ appear at _P_: in other words, it would displace it
towards the zenith. A star in the zenith is not displaced by refraction,
since the rays which reach the eye from it traverse the atmosphere
vertically. The farther a star is from the zenith, the more it is
displaced by refraction, since the greater is the obliquity with which
the rays from it enter the atmosphere.
[Illustration: Fig. 42.]
At the horizon the displacement by refraction is about half a degree;
but it varies considerably with the state of the atmosphere. Refraction
causes a heavenly body to appear above the horizon longer than it really
is above it, since it makes it appear to be on the horizon when it is
really half a degree below it.
The increase of refraction towards the horizon often makes the sun, when
near the horizon, appear distorted, the lower limb of the sun being
raised more than the upper limb. This distortion is shown in Fig. 43.
The vertical diameter of the sun appears to be considerably less than
the horizontal diameter.
[Illustration: Fig. 43.]
32. _Parallax._--_Parallax_ is the displacement of an object caused by a
change in the point of view from which it is seen. Thus in Fig. 44, the
top of the tower _S_ would be seen projected against the sky at _a_ by
an observer at _A_, and at _b_ by an observer at _B_. In passing from
_A_ to _B_, the top of the tower is displaced from _a_ to _b_, or by the
angle _aSb_. This angle is called the parallax of _S_, as seen from _B_
instead of _A_.
[Illustration: Fig. 44.]
The _geocentric parallax_ of a heavenly body is its displacement caused
by its being seen from the surface of the earth, instead of from the
centre of the earth. In Fig. 45, _R_ is the centre of the earth, and _O_
the point of observation on the surface of the earth. Stars at _S_,
_S'_, and _S''_, would, from the centre of the earth, appear at _Q_,
_Q'_, and _Q''_; while from the point _O_ on the surface of the earth,
these same stars would appear at _P_, _P'_ and _P''_, being displaced
from their position, as seen from the centre of the earth, by the angles
_QSP_, _Q'S'P'_, and _Q''S''P''_. It will be seen that parallax
displaces a body from the zenith towards the horizon, and that the
parallax of a body is greatest when it is on the horizon. The parallax
of a heavenly body when on the horizon is called its _horizontal
parallax_. A body in the zenith is not displaced by parallax, since it
would be seen in the same direction from both the centre and the surface
of the earth.
[Illustration: Fig. 45.]
The parallax of a body at _S'''_ is _Q'''S'''P_, which is seen to be
greater than _QSP_; that is to say, the parallax of a heavenly body
increases with its nearness to the earth. The distance and parallax of a
body are so related, that, either being known, the other may be
computed.
33. _Aberration._--_Aberration_ is a slight displacement of a star,
owing to an apparent change in the direction of the rays of light
which proceed from it, caused by the motion of the earth in its
orbit. If we walk rapidly in any direction in the rain, when the
drops are falling vertically, they will appear to come into our
faces from the direction in which we are walking. Our own motion has
apparently changed the direction in which the drops are falling.
[Illustration: Fig. 46.]
In Fig. 46 let _A_ be a gun of a battery, from which a shot is fired
at a ship, _DE_, that is passing. Let _ABC_ be the course of the
shot. The shot enters the ship's side at _B_, and passes out at the
other side at _C_; but in the mean time the ship has moved from _E_
to _e_, and the part _B_, where the shot entered, has been carried
to _b_. If a person on board the ship could see the ball as it
crossed the ship, he would see it cross in the diagonal line _bC_;
and he would at once say that the cannon was in the direction of
_Cb_. If the ship were moving in the opposite direction, he would
say that the cannon was just as far the other side of its true
position.
Now, we see a star in the direction in which the light coming from
the star appears to be moving. When we examine a star with a
telescope, we are in the same condition as the person who on
shipboard saw the cannon-ball cross the ship. The telescope is
carried along by the earth at the rate of eighteen miles a second:
hence the light will appear to pass through the tube in a slightly
different direction from that in which it is really moving: just as
the cannon-ball appears to pass through the ship in a different
direction from that in which it is really moving. Thus in Fig. 47, a
ray of light coming in the direction _SOT_ would appear to traverse
the tube _OT_ of a telescope, moving in the direction of the arrow,
as if it were coming in the direction _S'O_.
[Illustration: Fig. 47.]
As light moves with enormous velocity, it passes through the tube so
quickly, that it is apparently changed from its true direction only
by a very slight angle: but it is sufficient to displace the star.
This apparent change in the direction of light caused by the motion
of the earth is called _aberration of light_.
34. _The Planets._--On watching the stars attentively night after night,
it will be found, that while the majority of them appear _fixed_ on the
surface of the celestial sphere, so as to maintain their relative
positions, there are a few that _wander_ about among the stars,
alternately advancing towards the east, halting, and retrograding
towards the west. These wandering stars are called _planets_.
Their motions appear quite irregular; but, on the whole, their eastward
motion is in excess of their westward, and in a longer or shorter time
they all complete the circuit of the heavens. In almost every instance,
their paths are found to lie wholly in the belt of the zodiac.
[Illustration: Fig. 48.]
Fig. 48 shows a portion of the apparent path of one of the planets.
II.
THE SOLAR SYSTEM.
I. THEORY OF THE SOLAR SYSTEM.
35. _Members of the Solar System._--The solar system is composed of the
_sun_, _planets_, _moons_, _comets_, and _meteors_. Five planets,
besides the earth, are readily distinguished by the naked eye, and were
known to the ancients: these are _Mercury_, _Venus_, _Mars_, _Jupiter_,
and _Saturn_. These, with the _sun_ and _moon_, made up the _seven
planets_ of the ancients, from which the seven days of the week were
named.
The Ptolemaic System.
36. _The Crystalline Spheres._--We have seen that all the heavenly
bodies appear to be situated on the surface of the celestial sphere. The
ancients assumed that the stars were really fixed on the surface of a
crystalline sphere, and that they were carried around the earth daily by
the rotation of this sphere. They had, however, learned to distinguish
the planets from the stars, and they had come to the conclusion that
some of the planets were nearer the earth than others, and that all of
them were nearer the earth than the stars are. This led them to imagine
that the heavens were composed of a number of crystalline spheres, one
above another, each carrying one of the planets, and all revolving
around the earth from east to west, but at different rates. This
structure of the heavens is shown in section in Fig. 49.
[Illustration: Fig. 49.]
37. _Cycles and Epicycles._--The ancients had also noticed that, while
all the planets move around the heavens from west to east, their motion
is not one of uniform advancement. Mercury and Venus appear to oscillate
to and fro across the sun, while Jupiter and Saturn appear to oscillate
to and fro across a centre which is moving around the earth, so as to
describe a series of loops, as shown in Fig. 50.
[Illustration: Fig. 50.]
The ancients assumed that the planets moved in exact circles, and, in
fact, that all motion in the heavens was circular, the circle being the
simplest and most perfect curve. To account for the loops described by
the planets, they imagined that each planet revolved in a circle around
a centre, which, in turn, revolved in a circle around the earth. The
circle described by this centre around the earth they called the
_cycle_, and the circle described by the planet around this centre they
called the _epicycle_.
38. _The Eccentric._--The ancients assumed that the planets moved at a
uniform rate in describing the epicycle, and also the centre in
describing the cycle. They had, however, discovered that the planets
advance eastward more rapidly in some parts of their orbits than in
others. To account for this they assumed that the cycles described by
the centre, around which the planets revolved, were _eccentric_; that is
to say, that the earth was not at the centre of the cycle, but some
distance away from it, as shown in Fig. 51. _E_ is the position of the
earth, and _C_ is the centre of the cycle. The lines from _E_ are drawn
so as to intercept equal arcs of the cycle. It will be seen at once that
the angle between any pair of lines is greatest at _P_, and least at
_A_; so that, were a planet moving at the same rate at _P_ and _A_, it
would seem to be moving much faster at _P_. The point _P_ of the
planet's cycle was called its _perigee_, and the point A its _apogee_.
[Illustration: Fig. 51.]
As the apparent motion of the planets became more accurately known, it
was found necessary to make the system of cycles, epicycles, and
eccentrics exceedingly complicated to represent that motion.
The Copernican System.
39. _Copernicus._--Copernicus simplified the Ptolemaic system greatly by
assuming that the earth and all the planets revolved about the sun as a
centre. He, however, still maintained that all motion in the heavens was
circular, and hence he could not rid his system entirely of cycles and
epicycles.
Tycho Brahe's System.
40. _Tycho Brahe._--Tycho Brahe was the greatest of the early
astronomical observers. He, however, rejected the system of Copernicus,
and adopted one of his own, which was much more complicated. He held
that all the planets but the earth revolved around the sun, while the
sun and moon revolved around the earth. This system is shown in Fig. 52.
[Illustration: Fig. 52.]
Kepler's System.
41. _Kepler._--While Tycho Brahe devoted his life to the observation of
the planets. Kepler gave his to the study of Tycho's observations, for
the purpose of discovering the true laws of planetary motion. He
banished the complicated system of cycles, epicycles, and eccentrics
forever from the heavens, and discovered the three laws of planetary
motion which have rendered his name immortal.
42. _The Ellipse._--An _ellipse_ is a closed curve which has two points
within it, the sum of whose distances from every point on the curve is
the same. These two points are called the _foci_ of the ellipse.
[Illustration: Fig. 53.]
One method of describing an ellipse is shown in Fig. 53. Two tacks, _F_
and _F'_, are stuck into a piece of paper, and to these are fastened the
two ends of a string which is longer than the distance between the
tacks. A pencil is then placed against the string, and carried around,
as shown in the figure. The curve described by the pencil is an ellipse.
The two points _F_ and _F'_ are the foci of the ellipse: the sum of the
distances of these two points from every point on the curve is equal to
the length of the string. When half of the ellipse has been described,
the pencil must be held against the other side of the string in the same
way, and carried around as before.
The point _O_, half way between _F_ and _F'_, is called the _centre_ of
the ellipse; _AA'_ is the _major axis_ of the ellipse, and _CD_ is the
_minor axis_.
43. _The Eccentricity of the Ellipse._--The ratio of the distance
between the two foci to the major axis of the ellipse is called the
_eccentricity_ of the ellipse. The greater the distance between the two
foci, compared with the major axis of the ellipse, the greater is the
eccentricity of the ellipse; and the less the distance between the foci,
compared with the length of the major axis, the less the eccentricity of
the ellipse. The ellipse of Fig. 54 has an eccentricity of 1/8. This
ellipse scarcely differs in appearance from a circle. The ellipse of
Fig. 55 has an eccentricity of 1/2, and that of Fig. 56 an eccentricity
of 7/8.
[Illustration: Fig. 54.]
[Illustration: Fig. 55.]
[Illustration: Fig. 56.]
44. _Kepler's First Law._--Kepler first discovered that _all the planets
move from west to east in ellipses which have the sun as a common
focus_. This law of planetary motion is known as _Kepler's First Law_.
The planets appear to describe loops, because we view them from a moving
point.
The ellipses described by the planets differ in eccentricity; and,
though they all have one focus at the sun, their major axes have
different directions. The eccentricity of the planetary orbits is
comparatively small. The ellipse of Fig. 54 has seven times the
eccentricity of the earth's orbit, and twice that of the orbit of any of
the larger planets except Mercury; and its eccentricity is more than
half of that of the orbit of Mercury. Owing to their small eccentricity,
the orbits of the planets are usually represented by circles in
astronomical diagrams.
[Illustration: Fig. 57.]
45. _Kepler's Second Law._--Kepler next discovered that a planet's rate
of motion in the various parts of its orbit is such that _a line drawn
from the planet to the sun would always sweep over equal areas in equal
times_. Thus, in Fig. 57, suppose the planet would move from _P_ to
_P^1_ in the same time that it would move from _P^2_ to _P^3_, or from
_P^4_ to _P^5_; then the dark spaces, which would be swept over by a
line joining the sun and the planet, in these equal times, would all be
equal.
A line drawn from the sun to a planet is called the _radius vector_ of
the planet. The radius vector of a planet is shortest when the planet is
nearest the sun, or at _perihelion_, and longest when the planet is
farthest from the sun, or at _aphelion_: hence, in order to have the
areas equal, it follows that a planet must move fastest when at
perihelion, and slowest at aphelion.
_Kepler's Second Law_ of planetary motion is usually stated as follows:
_The radius vector of a planet describes equal areas in equal times in
every part of the planet's orbit_.
46. _Kepler's Third Law._--Kepler finally discovered that the periodic
times of the planets bear the following relation to the distances of the
planets from the sun: _The squares of the periodic times of the planets
are to each other as the cubes of their mean distances from the sun_.
This is known as _Kepler's Third Law_ of planetary motion. By _periodic
time_ is meant the time it takes a planet to revolve around the sun.
These three laws of Kepler's are the foundation of modern physical
astronomy.
The Newtonian System.
47. _Newton's Discovery._--Newton followed Kepler, and by means of his
three laws of planetary motion made his own immortal discovery of the
_law of gravitation_. This law is as follows: _Every portion of matter
in the universe attracts every other portion with a force varying
directly as the product of the masses acted upon, and inversely as the
square of the distances between them._
48. _The Conic Sections._--The _conic sections_ are the figures formed
by the various plane sections of a right cone. There are four classes of
figures formed by these sections, according to the angle which the plane
of the section makes with the axis of the cone.
_OPQ_, Fig. 58, is a right cone, and _ON_ is its axis. Any section,
_AB_, of this cone, whose plane is perpendicular to the axis of the
cone, is a _circle_.
[Illustration: Fig. 58.]
Any section, _CD_, of this cone, whose plane is oblique to the axis, but
forms with it an angle greater than _NOP_, is an _ellipse_. The less the
angle which the plane of the section makes with the axis, the more
elongated is the ellipse.
Any section, _EF_, of this cone, whose plane makes with the axis an
angle equal to _NOP_, is a _parabola_. It will be seen, that, by
changing the obliquity of the plane _CD_ to the axis _NO_, we may pass
uninterruptedly from the circle through ellipses of greater and greater
elongation to the parabola.
Any section, _GH_, of this cone, whose plane makes with the axis _ON_ an
angle less than _NOP_, is a _hyperbola_.
[Illustration: Fig. 59.]
It will be seen from Fig. 59, in which comparative views of the four
conic sections are given, that the circle and the ellipse are _closed_
curves, or curves which return into themselves. The parabola and the
hyperbola are, on the contrary, _open_ curves, or curves which do not
return into themselves.
49. _A Revolving Body is continually Falling towards its Centre of
Revolution._--In Fig. 60 let _M_ represent the moon, and _E_ the
earth around which the moon is revolving in the direction _MN_. It
will be seen that the moon, in moving from M to N, falls towards the
earth a distance equal to _mN_. It is kept from falling into the
earth by its orbital motion.
[Illustration: Fig. 60.]
The fact that a body might be projected forward fast enough to keep
it from falling into the earth is evident from Fig. 61. _AB_
represents the level surface of the ocean, _C_ a mountain from the
summit of which a cannon-ball is supposed to be fired in the
direction _CE_. _AD_ is a line parallel with _CE_; _DB_ is a line
equal to the distance between the two parallel lines _AD_ and _CE_.
This distance is equal to that over which gravity would pull a ball
towards the centre of the earth in a minute. No matter, then, with
what velocity the ball _C_ is fired, at the end of a minute it will
be somewhere on the line _AD_. Suppose it were fired fast enough to
reach the point _D_ in a minute: it would be on the line _AD_ at the
end of the minute, but still just as far from the surface of the
water as when it started. It will be seen, that, although it has all
the while been falling towards the earth, it has all the while kept
at exactly the same distance from the surface. The same thing would
of course be true during each succeeding minute, till the ball came
round to _C_ again, and the ball would continue to revolve in a
circle around the earth.
[Illustration: Fig. 61.]
50. _The Form of a Body's Orbit depends upon the Rate of its Forward
Motion._--If the ball _C_ were fired fast enough to reach the line _AD_
beyond the point _D_, it would be farther from the surface at the end of
the second than when it started. Its orbit would no longer be circular,
but _elliptical_. If the speed of projection were gradually augmented,
the orbit would become a more and more elongated ellipse. At a certain
rate of projection, the orbit would become a _parabola_; at a still
greater rate, a _hyperbola_.
51. _The Moon held in her Orbit by Gravity._--Newton compared the
distance _mN_ that the moon is drawn to the earth in a given time, with
the distance a body near the surface of the earth would be pulled toward
the earth in the same time; and he found that these distances are to
each other inversely as the squares of the distances of the two bodies
from the centre of the earth. He therefore concluded that _the moon is
drawn to the earth by gravity_, and that the _intensity of gravity
decreases as the square of the distance increases_.
[Illustration: Fig. 62.]
52. _Any Body whose Orbit is a Conic Section, and which moves according
to Kepler's Second Law, is acted upon by a Force varying inversely as
the Square of the Distance._--Newton compared the distance which any
body, moving in an ellipse, according to Kepler's Second Law, is drawn
towards the sun in the same time in different parts of its orbit. He
found these distances in all cases to vary inversely as the square of
the distance of the planet from the sun. Thus, in Fig. 62, suppose a
planet would move from _K_ to _B_ in the same time that it would move
from _k_ to _b_ in another part of its orbit. In the first instance the
planet would be drawn towards the sun the distance _AB_, and in the
second instance the distance _ab_. Newton found that _AB : ab = (SK)^2 :
(Sk)^2_. He also found that the same would be true when the body moved
in a parabola or a hyperbola: hence he concluded that _every body that
moves around the sun in an ellipse, a parabola, or a hyperbola, is
moving under the influence of gravity_.
[Transcriber's Note: In Newton's equation above, (SK)^2 means to group S
and K together and square their product. In the original book, instead
of using parentheses, there was a vinculum, a horizontal bar, drawn over
the S and the K to express the same grouping.]
[Illustration: Fig. 63.]
53. _The Force that draws the Different Planets to the Sun Varies
inversely as the Squares of the Distances of the Planets from the
Sun._--Newton compared the distances _jK_ and _eF_, over which two
planets are drawn towards the sun in the same time, and found these
distances to vary inversely as the squares of the distances of the
planets from the sun: hence he concluded that _all the planets are held
in their orbits by gravity_. He also showed that this would be true of
any two bodies that were revolving around the sun's centre, according to
Kepler's Third Law.
54. _The Copernican System._--The theory of the solar system which
originated with Copernicus, and which was developed and completed by
Kepler and Newton, is commonly known as the _Copernican System_. This
system is shown in Fig. 64.
[Illustration: Fig. 64.]
II. THE SUN AND PLANETS.
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.
[Illustration: Fig. 65.]
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.
[Illustration: Fig. 66.]
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.
[Illustration: Fig. 67.]
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_.