changes in the sun's apparent diameter_, and, secondly, _by variations
in his angular velocity_. The same object appears to us smaller in
proportion as it is more distant; and if we see a heavenly body varying
in size, at different times, we infer that it is at different distances
from us; that when largest, it is nearest to us, and when smallest,
furthest off. Now, when the sun's diameter is accurately measured by
instruments, it is found to vary from day to day; being, when greatest,
more than thirty-two minutes and a half, and when smallest, only
thirty-one minutes and a half,--differing, in all, about seventy-five
seconds. When the diameter is greatest, which happens in January, we
know that the sun is nearest to us; and when the diameter is least,
which occurs in July, we infer that the sun is at the greatest distance
from us. The point where the earth, or any planet, in its revolution, is
nearest the sun, is called its _perihelion_; the point where it is
furthest from the sun, its _aphelion_. Suppose, then, that, about the
first of January, when the diameter of the sun is greatest, we draw a
line, E _a_, Fig. 32, to represent it, and afterwards, every ten days,
draw other lines, E _b_, E _c_, &c.; increasing in the same ratio as the
apparent diameters of the sun decrease. These lines must be drawn at
such a distance from each other, that the triangles, E _a b_, E _b c_,
&c., shall be all equal to each other, for a reason that will be
explained hereafter. On connecting the extremities of these lines, we
shall obtain the figure of the earth's orbit.
Similar conclusions may be drawn from observations on the sun's _angular
velocity_. A body appears to move most rapidly when nearest to us.
Indeed, the apparent velocity increases rapidly, as it approaches us,
and as rapidly diminishes, when it recedes from us. If it comes twice as
near as before, it appears to move not merely twice as swiftly, but four
times as swiftly; if it comes ten times nearer, its apparent velocity
is one hundred times as great as before. We say, therefore, that the
velocity varies inversely as the square of the distance; for, as the
distance is diminished ten times, the velocity is increased the square
of ten; that is, one hundred times. Now, by noting the time it takes the
sun, from day to day, to cross the central wire of the
transit-instrument, we learn the comparative velocities with which it
moves at different times; and from these we derive the comparative
distances of the sun at the corresponding times; and laying down these
relative distances in a diagram, as before, we get our first notions of
the actual figure of the earth's orbit, or the path which it describes
in its annual revolution around the sun.
Having now learned the fact, that the earth moves around the sun, not in
a circular but in an elliptical orbit, you will desire to know by what
forces it is impelled, to make it describe this figure, with such
uniformity and constancy, from age to age. It is commonly said, that
gravity causes the earth and the planets to circulate around the sun;
and it is true that it is gravity which turns them aside from the
straight line in which, by the first law of motion, they tend to move,
and thus causes them to revolve around the sun. But what force is that
which gave to them this original impulse, and impressed upon them such a
tendency to move forward in a straight line? The name _projectile_ force
is given to it, because it is the same _as though_ the earth were
originally projected into space, when first created; and therefore its
motion is the result of two forces, the projectile force, which would
cause it to move forward in a straight line which is a tangent to its
orbit, and gravitation, which bends it towards the sun. But before you
can clearly understand the nature of this motion, and the action of the
two forces that produce it, I must explain to you a few elementary
principles upon which this and all the other planetary motions depend.
You have already learned, that when a body is acted on by two forces, in
different directions, it moves in the direction of neither, but in some
direction between them. If I throw a stone horizontally, the attraction
of the earth will continually draw it downward, out of the line of
direction in which it was thrown, and make it descend to the earth in a
curve. The particular form of the curve will depend on the velocity with
which it is thrown. It will always _begin_ to move in the line of
direction in which it is projected; but it will soon be turned from that
line towards the earth. It will, however, continue nearer to the line of
projection in proportion as the velocity of projection is greater. Thus,
let A C, Fig. 33, be perpendicular to the horizon, and A B parallel to
it, and let a stone be thrown from A, in the direction of A B. It will,
in every case, commence its motion in the line A B, which will therefore
be a tangent to the curve it describes; but, if it is thrown with a
small velocity, it will soon depart from the tangent, describing the
line A D; with a greater velocity, it will describe a curve nearer the
tangent, as A E; and with a still greater velocity, it will describe the
curve A F.
[Illustration Fig. 33.]
As an example of a body revolving in an orbit under the influence of two
forces, suppose a body placed at any point, P, Fig. 34, above the
surface of the earth, and let P A be the direction of the earth's
centre; that is, a line perpendicular to the horizon. If the body were
allowed to move, without receiving any impulse, it would descend to the
earth in the direction P A with an accelerated motion. But suppose that,
at the moment of its departure from P, it receives a blow in the
direction P B, which would carry it to B in the time the body would fall
from P to A; then, under the influence of both forces, it would descend
along the curve P D. If a stronger blow were given to it in the
direction P B, it would describe a larger curve, P E; or, finally, if
the impulse were sufficiently strong, it would circulate quite around
the earth, and return again to P, describing the circle P F G. With a
velocity of projection still greater, it would describe an ellipse, P I
K; and if the velocity be increased to a certain degree, the figure
becomes a parabola, L P M,--a curve which never returns into itself.
[Illustration Fig. 34.]
In Fig. 35, page 154, suppose the planet to have passed the point C, at
the aphelion, with so small a velocity, that the attraction of the sun
bends its path very much, and causes it immediately to begin to approach
towards the sun. The sun's attraction will increase its velocity, as it
moves through D, E, and F, for the sun's attractive force on the planet,
when at D, is acting in the direction D S; and, on account of the small
angle made between D E and D S, the force acting in the line D S helps
the planet forward in the path D E, and thus increases its velocity. In
like manner, the velocity of the planet will be continually increasing
as it passes through D, E, and F; and though the attractive force, on
account of the planet's nearness, is so much increased, and tends,
therefore, to make the orbit more curved, yet the velocity is also so
much increased, that the orbit is not more curved than before; for the
same increase of velocity, occasioned by the planet's approach to the
sun, produces a greater increase of centrifugal force, which carries it
off again. We may see, also, the reason why, when the planet has reached
the most distant parts of its orbit, it does not entirely fly off, and
never return to the sun; for, when the planet passes along H, K, A, the
sun's attraction retards the planet, just as gravity retards a ball
rolled up hill; and when it has reached C, its velocity is very small,
and the attraction to the centre of force causes a great deflection from
the tangent, sufficient to give its orbit a great curvature, and the
planet wheels about, returns to the sun, and goes over the same orbit
again. As the planet recedes from the sun, its centrifugal force
diminishes faster than the force of gravity, so that the latter finally
preponderates.
[Illustration Fig. 35.]
I shall conclude what I have to say at present, respecting the motion of
the earth around the sun, by adding a few words respecting the
precession of the equinoxes.
The _precession of the equinoxes_ is a slow but continual shifting of
the equinoctial points, from east to west. Suppose that we mark the
exact place in the heavens where, during the present year, the sun
crosses the equator, and that this point is close to a certain star;
next year, the sun will cross the equator a little way westward of that
star, and so every year, a little further westward, until, in a long
course of ages, the place of the equinox will occupy successively every
part of the ecliptic, until we come round to the same star again. As,
therefore, the sun revolving from west to east, in his apparent orbit,
comes round to the point where it left the equinox, it meets the equinox
before it reaches that point. The appearance is as though the equinox
_goes forward_ to meet the sun, and hence the phenomenon is called the
_precession_ of the equinoxes; and the fact is expressed by saying, that
the equinoxes retrograde on the ecliptic, until the line of the
equinoxes (a straight line drawn from one equinox to the other) makes a
complete revolution, from east to west. This is of course a retrograde
motion, since it is contrary to the order of the signs. The equator is
conceived as _sliding_ westward on the ecliptic, always preserving the
same inclination to it, as a ring, placed at a small angle with another
of nearly the same size which remains fixed, may be slid quite around
it, giving a corresponding motion to the two points of intersection. It
must be observed, however, that this mode of conceiving of the
precession of the equinoxes is purely imaginary, and is employed merely
for the convenience of representation.
The amount of precession annually is fifty seconds and one tenth;
whence, since there are thirty-six hundred seconds in a degree, and
three hundred and sixty degrees in the whole circumference of the
ecliptic, and consequently one million two hundred and ninety-six
thousand seconds, this sum, divided by fifty seconds and one tenth,
gives twenty-five thousand eight hundred and sixty-eight years for the
period of a complete revolution of the equinoxes.
Suppose we now fix to the centre of each of the two rings, before
mentioned, a wire representing its axis, one corresponding to the axis
of the ecliptic, the other to that of the equator, the extremity of each
being the pole of its circle. As the ring denoting the equator turns
round on the ecliptic, which, with its axis, remains fixed, it is easy
to conceive that the axis of the equator revolves around that of the
ecliptic, and the pole of the equator around the pole of the ecliptic,
and constantly at a distance equal to the inclination of the two
circles. To transfer our conceptions to the celestial sphere, we may
easily see that the axis of the diurnal sphere (that of the earth
produced) would not have its pole constantly in the same place among the
stars, but that this pole would perform a slow revolution around the
pole of the ecliptic, from east to west, completing the circuit in about
twenty-six thousand years. Hence the star which we now call the
pole-star has not always enjoyed that distinction, nor will it always
enjoy it, hereafter. When the earliest catalogues of the stars were
made, this star was twelve degrees from the pole. It is now one degree
twenty-four minutes, and will approach still nearer; or, to speak more
accurately, the pole will come still nearer to this star, after which it
will leave it, and successively pass by others. In about thirteen
thousand years, the bright star Lyra (which lies near the circle in
which the pole of the equator revolves about the pole of the ecliptic,
on the side opposite to the present pole-star) will be within five
degrees of the pole, and will constitute the pole-star. As Lyra now
passes near our zenith, you might suppose that the change of position of
the pole among the stars would be attended with a change of altitude of
the north pole above the horizon. This mistaken idea is one of the many
misapprehensions which result from the habit of considering the horizon
as a fixed circle in space. However the pole might shift its position in
space, we should still be at the same distance from it, and our horizon
would always reach the same distance beyond it.
The time occupied by the sun, in passing from the equinoctial point
round to the same point again, is called the _tropical year_. As the sun
does not perform a complete revolution in this interval, but falls short
of it fifty seconds and one tenth, the tropical year is shorter than the
sidereal by twenty minutes and twenty seconds, in mean solar time, this
being the time of describing an arc of fifty seconds and one tenth, in
the annual revolution.
The changes produced by the precession of the equinoxes, in the apparent
places of the circumpolar stars, have led to some interesting results in
_chronology_. In consequence of the retrograde motion of the equinoctial
points, the _signs_ of the ecliptic do not correspond, at present, to
the _constellations_ which bear the same names, but lie about one sign,
or thirty degrees, westward of them. Thus, that division of the ecliptic
which is called the sign Taurus lies in the constellation Aries, and the
sign Gemini, in the constellation Taurus. Undoubtedly, however, when the
ecliptic was thus first divided, and the divisions named, the several
constellations lay in the respective divisions which bear their names.
LETTER XV.
THE MOON.
"Soon as the evening shades prevail
The Moon takes up the wondrous tale,
And nightly to the listening earth
Repeats the story of her birth."--_Addison._
HAVING now learned so much of astronomy as relates to the earth and the
sun, and the mutual relations which exist between them, you are prepared
to enter with advantage upon the survey of the other bodies that compose
the solar system. This being done, we shall then have still before us
the boundless range of the fixed stars.
The moon, which next claims our notice, has been studied by astronomers
with greater attention than any other of the heavenly bodies, since her
comparative nearness to the earth brings her peculiarly within the range
of our telescopes, and her periodical changes and very irregular
motions, afford curious subjects, both for observation and speculation.
The mild light of the moon also invites our gaze, while her varying
aspects serve barbarous tribes, especially, for a kind of dial-plate
inscribed on the face of the sky, for weeks, and months, and times, and
seasons.
The moon is distant from the earth about two hundred and forty thousand
miles; or, more exactly, two hundred and thirty-eight thousand five
hundred and forty-five miles. Her angular or apparent diameter is about
half a degree, and her real diameter, two thousand one hundred and sixty
miles. She is a companion, or satellite, to the earth, revolving around
it every month, and accompanying us in our annual revolution around the
sun. Although her nearness to us makes her appear as a large and
conspicuous object in the heavens, yet, in comparison with most of the
other celestial bodies, she is in fact very small, being only one
forty-ninth part as large as the earth, and only about one seventy
millionth part as large as the sun.
The moon shines by light borrowed from the sun, being itself an opaque
body, like the earth. When the disk, or any portion of it, is
illuminated, we can plainly discern, even with the naked eye, varieties
of light and shade, indicating inequalities of surface which we imagine
to be land and water. I believe it is the common impression, that the
darker portions are land and the lighter portions water; but if either
part is water, it must be the darker regions. A smooth polished surface,
like water, would reflect the sun's light like a mirror. It would, like
a convex mirror, form a diminished image of the sun, but would not
itself appear luminous like an uneven surface, which multiplies the
light by numerous reflections within itself. Thus, from this cause, high
broken mountainous districts appear more luminous than extensive plains.
[Illustration Figures 36, 37. TELESCOPIC VIEWS OF THE MOON.]
By the aid of the telescope, we may see undoubted indications of
mountains and valleys. Indeed, with a good glass, we can discover the
most decisive evidence that the surface of the moon is exceedingly
varied,--one part ascending in lofty peaks, another clustering in
huge mountain groups, or long ranges, and another bearing all the marks
of deep caverns or valleys. You will not, indeed, at the first sight of
the moon through a telescope, recognise all these different objects. If
you look at the moon when half her disk is enlightened, (which is the
best time for seeing her varieties of surface,) you will, at the first
glance, observe a motley appearance, particularly along the line called
the _terminator_, which separates the enlightened from the unenlightened
part of the disk. (Fig. 37.) On one side of the terminator, within the
dark part of the disk, you will see illuminated points, and short,
crooked lines, like rude characters marked with chalk on a black ground.
On the other side of the terminator you will see a succession of little
circular groups, appearing like numerous bubbles of oil on the surface
of water. The further you carry your eye from the terminator, on the
same side of it, the more indistinctly formed these bubbles appear,
until towards the edge of the moon they assume quite a different aspect.
Some persons, when they look into a telescope for the first time, having
heard that mountains and valleys are to be seen, and discovering nothing
but these unmeaning figures, break off in disappointment, and have their
faith in these things rather diminished than increased. I would advise
you, therefore, before you take even your first view of the moon through
a telescope, to form as clear an idea as you can, how mountains, and
valleys, and caverns, situated at such a distance from the eye, ought to
look, and by what marks they may be recognised. Seize, if possible, the
most favorable period, (about the time of the first quarter,) and
previously learn from drawings and explanations, how to interpret every
thing you see.
What, then, ought to be the respective appearances of mountains,
valleys, and deep craters, or caverns, in the moon? The sun shines on
the moon in the same way as it shines on the earth; and let, us reflect,
then, upon the manner in which it strikes similar objects here. One
half the globe is constantly enlightened; and, by the revolution of the
earth on its axis, the terminator, or the line which separates the
enlightened from the unenlightened part of the earth, travels along from
east to west, over different places, as we see the moon's terminator
travel over her disk from new to full moon; although, in the case of the
earth, the motion is more rapid, and depends on a different cause. In
the morning, the sun's light first strikes upon the tops of the
mountains, and, if they are very high, they may be brightly illuminated
while it is yet night in the valleys below. By degrees, as the sun
rises, the circle of illumination travels down the mountain, until at
length it reaches the bottom of the valleys; and these in turn enjoy the
full light of day. Again, a mountain casts a shadow opposite to the sun,
which is very long when the sun first rises, and shortens continually as
the sun ascends, its length at a given time, however, being proportioned
to the height of the mountain; so that, if the shadow be still very long
when the sun is far above the horizon, we infer that the mountain is
very lofty. We may, moreover, form some judgment of the shape of a
mountain, by observing that of its shadow.
Now, the moon is so distant that we could not easily distinguish places
simply by their elevations, since they would be projected into the same
imaginary plane which constitutes the apparent disk of the moon; but the
foregoing considerations would enable us to infer their existence. Thus,
when you view the moon at any time within her first quarter, but better
near the end of that period, you will observe, on the side of the
terminator within the dark part of the disk, the tops of mountains which
the light of the sun is just striking, as the morning sun strikes the
tops of mountains on the earth. These you will recognise by those white
specks and little crooked lines, before mentioned, as is represented in
Fig. 37. These bright points and lines you will see altering their
figure, every hour, as they come more and more into the sun's light;
and, mean-while, other bright points, very minute at first, will start
into view, which also in turn grow larger as the terminator approaches
them, until they fall into the enlightened part of the disk. As they
fall further and further within this part, you will have additional
proofs that they are mountains, from the shadows which they cast on the
plain, always in a direction opposite to the sun. The mountain itself
may entirely disappear, or become confounded with the other enlightened
portions of the surface; but its position and its shape may still be
recognised by the dark line which it projects on the plane. This line
will correspond in shape to that of the mountain, presenting at one time
a long serpentine stripe of black, denoting that the mountain is a
continued range; at another time exhibiting a conical figure tapering to
a point, or a series of such sharp points; or a serrated, uneven
termination, indicating, in each case respectively, a conical mountain,
or a group of peaks, or a range with lofty cliffs. All these appearances
will indeed be seen in miniature; but a little familiarity with them
will enable you to give them, in imagination, their proper dimensions,
as you give to the pictures of known animals their due sizes, although
drawn on a scale far below that of real life.