points of which have contemporaneously the same noon. Were the lengths
and curvatures of different meridians known, the figure of the earth
might be determined; but the length of one degree is sufficient to give
the figure of the earth, if it be measured on different meridians, and
in a variety of latitudes; for if the earth were a sphere, all degrees
would be of the same length, but if not, the lengths of the degrees will
be greatest where the curvature is least; a comparison of the length of
the degrees in different parts of the earth's surface will therefore
determine its size and form.
An arc of the meridian may be measured by observing the latitude of its
extreme points, and then measuring the distance between them in feet or
fathoms; the distance thus determined on the surface of the earth,
divided by the degrees and parts of a degree contained in the difference
of the latitudes, will give the exact length of one degree, the
difference of the latitudes being the angle contained between the
verticals at the extremities of the arc. This would be easily
accomplished were the distance unobstructed, and on a level with the
sea; but on account of the innumerable obstacles on the surface of the
earth, it is necessary to connect the extreme points of the arc by a
series of triangles, the sides and angles of which are either measured
or computed, so that the length of the arc is ascertained with much
laborious computation. In consequence of the inequalities of the
surface, each triangle is in a different plane; they must therefore be
reduced by computation to what they would have been, had they been
measured on the surface of the sea; and as the earth is spherical, they
require a correction to reduce them from plane to spherical triangles.
Arcs of the meridian have been measured in a variety of latitudes, both
north and south, as well as arcs perpendicular to the meridian. From
these measurements it appears that the length of the degrees increase
from the equator to the poles, nearly as the square of the sine of the
latitude; consequently, the convexity of the earth diminishes from the
equator to the poles. Many discrepancies occur, but the figure that most
nearly follows this law is an ellipsoid of revolution, whose equatorial
radius is 3962.6 miles, and the polar radius 3949.7; the difference, or
12.9 miles, divided by the equatorial radius, is 1/308·7, or 1/309
nearly; this fraction is called the compression of the earth, because,
according as it is greater or less, the terrestrial ellipsoid is more
or less flattened at the poles; it does not differ much from that given
by the lunar inequalities. If we assume the earth to be a sphere, the
length of a degree of the meridian is 69 1/22 British miles; therefore
360 degrees, or the whole circumference of the globe is 24856, and the
diameter, which is something less than a third of the circumference, is
7916 or 8000 miles nearly. Eratosthenes, who died 194 years before the
Christian era, was the first to give an approximate value of the earth's
circumference, by the mensuration of an arc between Alexandria and Syene.
But there is another method of finding the figure of the earth, totally
independent of either of the preceding. If the earth were a homogeneous
sphere without rotation, its attraction on bodies at its surface would
be everywhere the same; if it be elliptical, the force of gravity
theoretically ought to increase, from the equator to the pole as the
square of the sine of the latitude; but for a spheroid in rotation, by
the laws of mechanics the centrifugal force varies as the square of the
sine of the latitude from the equator where it is greatest, to the pole
where it vanishes; and as it tends to make bodies fly off the surface,
it diminishes the effects of gravity by a small quantity. Hence by
gravitation, which is the difference of these two forces, the fall of
bodies ought to be accelerated in going from the equator to the poles,
proportionably to the square of the sine of the latitude; and the weight
of the same body ought to increase in that ratio. This is directly proved
by the oscillations of the pendulum; for if the fall of bodies be
accelerated, the oscillations will be more rapid; and that they may
always be performed in the same time, the length of the pendulum must
be altered. Now, by numerous and very careful experiments, it is proved
that a pendulum, which makes 86400 oscillations in a mean day at the
equator, will do the same at every point of the earth's surface, if
its length be increased in going to the pole, as the square of the
sine of the latitude. From the mean of these it appears that the
compression of the terrestrial spheroid is about 1/342, which does not
differ much from that given by the lunar inequalities, and from the arcs
of the meridian. The near coincidence of these three values, deduced
by methods so entirely independent of each other, shows that the mutual
tendencies of the centres of the celestial bodies to one another, and
the attraction of the earth for bodies at its surface, result from the
reciprocal attraction of all their particles. Another proof may be added;
the nutation of the earth's axis, and the precession of the equinoxes,
are occasioned by the action of the sun and moon on the protuberant
matter at the earth's equator; and although these inequalities do not
give the absolute value of the terrestrial compression, they show that
the fraction expressing it is comprised between the limits
1/279 and 1/578.
It might be expected that the same compression should result from each,
if the different methods of observation could be made without error.
This, however, is not the case; for such discrepancies are found both
in the degrees of the meridian and in the length of the pendulum, as
show that the figure of the earth is very complicated; but they are
so small when compared with the general results, that they may be
disregarded. The compression deduced from the mean of the whole,
appears to be about 1/320; that given by the lunar theory has the advantage
of being independent of the irregularities at the earth's surface,
and of local attractions. The form and size of the earth being determined,
it furnishes a standard of measure with which the dimensions of the
solar system may be compared.
The parallax of a celestial body is the angle under which the radius
of the earth would be seen if viewed from the centre of that body;
it affords the means of ascertaining the distances of the sun, moon,
and planets. Suppose that, when the moon is in the horizon at the
instant of rising or setting, lines were drawn from her centre to the
spectator and to the centre of the earth, these would form a right-angled
triangle with the terrestrial radius, which is of a known length;
and as the parallax or angle at the moon can be measured, all the angles
and one side are given; whence the distance of the moon from the centre
of the earth may be computed. The parallax of an object may be found,
if two observers under the same meridian, but at a very great distance
from one another, observe its zenith distances on the same day at the
time of its passage over the meridian. By such contemporaneous
observations at the Cape of Good Hope and at Berlin, the mean horizontal
parallax of the moon was found to be 3454"·2; whence the mean distance
of the moon is about sixty times the mean terrestrial radius, or 240000
miles nearly. Since the parallax is equal to the radius of the earth
divided by the distance of the moon; under the same parallel of latitude
it varies with the distance of the moon from the earth, and proves the
ellipticity of the lunar orbit; and when the moon is at her mean
distance, it varies with the terrestrial radii, thus showing that the
earth is not a sphere.
Although the method described is sufficiently accurate for finding the
parallax of an object so near as the moon, it will not answer for the
sun which is so remote, that the smallest error in observation would
lead to a false result; but by the transits of Venus that difficulty
is obviated. When that planet is in her nodes, or within 1 1/4° of them,
that is, in, or nearly in the plane of the ecliptic, she is occasionally
seen to pass over the sun like a block spot. If we could imagine that
the sun and Venus had no parallax, the line described by the planet on
his disc, and the duration of the transit, would be the same to all
the inhabitants of the earth; but as the sun is not so remote but that
the semidiameter of the earth has a sensible magnitude when viewed from
his centre, the line described by the planet in its passage over his
disc appears to be nearer to his centre or farther from it, according
to the position of the observer; So that the duration of the transit
varies with the different points of the earth's surface at which it is
observed. This difference of time, being entirely the effect of parallax,
furnishes the means of computing it from the known motions of the earth
and Venus, by the same method as for the eclipses of the sun. In fact
the ratio of the distances of Venus and the sun from the earth at the
time of the transit, are known from the theory of their elliptical
motion; consequently, the ratio of the parallaxes of these two bodies,
being inversely as their distances, is given; and as the transit gives
the difference of the parallaxes, that of the sun is obtained. In 1769,
the parallax of the sun was determined by observations of a transit of
Venus made at Wardhus in Lapland, and at Otaheite in the South Sea,
the latter observation being the object of Cook's first voyage. The
transit lasted about six hours at Otaheite, and the difference in the
duration at these two stations was eight minutes; whence the sun's
parallax was found to be 8"·72; but by other considerations it has
subsequently been reduced to 8"·575; from which the mean distance of
the sun appears to be about 95996000, or ninety-six millions of miles
nearly. This is confirmed by an inequality in the motion of the moon,
which depends on the parallax of the sun, and which when compared
with observation gives 8"·6 for the sun's parallax.
The parallax of Venus is determined by her transits, that of Mars
by direct observation. The distances of these two planets from the
earth are therefore known in terrestrial radii; consequently their
mean distances from the sun may be computed and as the ratios of the
distances of the planets from the sun are known by Kepler's law,
their absolute distances in miles are easily found.
Far as the earth seems to be from the sun, it is near to him when
compared with Uranus; that planet is no less than 1843 millions of
miles from the luminary that warms and enlivens the world; to it,
situate on the verge of the system, the sun must appear not much
larger than Venus does to us. The earth cannot even be visible as a
telescopic object to a body so remote; yet man, the inhabitant of the
earth, soars beyond the vast dimensions of the system to which his
planet belongs, and assumes the diameter of its orbit as the base
of a triangle, whose apex extends to the stars.
Sublime as the idea is, this assumption proves ineffectual, for
the apparent places of the fixed stars are not sensibly changed by
the earth's annual revolution; and with the aid derived from the
refinements of modern astronomy and the most perfect instruments,
it is still a matter of doubt whether a sensible parallax has been
detected, even in the nearest of these remote suns. If a fixed star
had the parallax of one second, its distance from the sun would be
20500000 millions of miles. At such a distance not only the terrestrial
orbit shrinks to a point, but, where the whole solar system, when
seen in the focus of the most powerful telescope, might be covered
by the thickness of a spider's thread. Light, flying at the rate
of 200000 miles in a second, would take three years and seven days
to travel over that space; one of the nearest stars may therefore
have been kindled or extinguished more than three years before we
could have been aware of so mighty an event. But this distance must
be small when compared with that of the most remote of the bodies
which are visible in the heavens. The fixed stars are undoubtedly
luminous like the sun; it is therefore probable that they are not
nearer to one another than the sun is to the nearest of them. In
the milky way and the other starry nebulæ, some of the stars that
seem to us to be close to others, may be far behind them in the
boundless depth of space; nay, may rationally be supposed to be
situated many thousand times further off: light would therefore
require thousands of years to come to the earth from those myriads
of suns, of which our own is but 'the dim and remote companion.'
The masses of such planets as have no satellites are known by comparing
the inequalities they produce in the motions of the earth and of each
other, determined theoretically, with the same inequalities given by
observation, for the disturbing cause must necessarily be proportional
to the effect it produces. But as the quantities of matter in any two
primary planets are directly as the cubes of the mean distances at which
their satellites revolve, and inversely as the squares of their periodic
times, the mass of the sun and of any planets which have satellites, may
be compared with the mass of the earth. In this manner it is computed
that the mass of the sun is 354936 times greater than that of the earth;
whence the great perturbations of the moon and the rapid motion of the
perigee and nodes of her orbit. Even Jupiter, the largest of the
planets, is 1070.5 times less than the sun. The mass of the moon is
determined from four different sources,--from her action on the
terrestrial equator, which occasions the rotation in the axis of
rotation; from her horizontal parallax, from an inequality she produces
in the sun's longitude, and from her action on the titles. The three
first quantities, computed from theory, and compared with their observed
values, give her mass respectively equal to the 1/71, 1/74·2, and 1/69·2
part of that of the earth, which do not differ very much from each
other; but, from her action in raising the tides, which furnishes
the fourth method, her mass appears to be about the seventy-fifth part
of that of the earth, a value that cannot differ much from the truth.
The apparent diameters of the sun, moon, and planets are determined by
measurement; therefore their real diameters may be compared with that of
the earth; for the real diameter of a planet is to the real diameter of
the earth, or 8000 miles, as the apparent diameter of the planet to the
apparent diameter of the earth as seen from the planet, that is, to
twice the parallax of the planet The mean apparent diameter of the sun
is 1920", and with the solar parallax 8"·65, it will be found that
the diameter of the sun is about 888000 miles; therefore, the centre of
the sun were to coincide with the centre of the earth, his volume would
not only include the orbit of the moon, but would extend nearly as far
again, for the moon's mean distance from the earth is about sixty times
the earth's mean radius or 240000 miles; so that twice the distance of
the moon is 480000 miles, which differs but little from the solar
radius; his equatorial radius is probably not much less than the major
axis of the lunar orbit.
The diameter of the moon is only 2160 miles; and Jupiter's diameter of
88000 miles is incomparably less than that of the sun The diameter of
Pallas does not much exceed 71 miles, so that an inhabitant of that
planet, in one of our steam-carriages, might go round his world in five
or six hours.
The oblate form of the celestial bodies indicates rotatory motion, and
this has been confirmed, in most cases, by tracing spots on their
surfaces, whence their poles and times of rotation have been determined.
The rotation of Mercury is unknown, on account of his proximity to the
sun; and that of the new planets has not yet been ascertained. The sun
revolves in twenty-five days ten hours, about an axis that is directed
towards a point half way between the pole star and Lyra, the plane of
rotation being inclined a little more than 70° to that on which the
earth revolves. From the rotation of the sun, there is every reason to
believe that he has a progressive motion in space, although the
direction to which he tends is as yet unknown; but in consequence of the
reaction of the planets, he describes a small irregular orbit about the
centre of inertia of the system, never deviating from his position by
more than twice his own diameter, or about seven times the distance of
the moon from the earth.
The sun and all his attendants rotate from west to east on axes that
remain nearly parallel to themselves in every point of their orbit, and
with angular velocities that are sensibly uniform. Although the
uniformity in the direction of their rotation is a circumstance hitherto
unaccounted for in the economy of Nature, yet from the design and
adaptation of every other part to the perfection of the whole, a
coincidence so remarkable cannot be accidental; and as the revolutions
of the planets and satellites are also from west to east, it is evident
that both must have arisen from the primitive causes which have
determined the planetary motions.
The larger planets rotate in shorter periods than the smaller planets
and the earth; their compression is consequently greater, and the action
of the sun and of their satellites occasions a nutation in their axes,
and a precession of their equinoxes, similar to that which obtains in
the terrestrial spheroid from the attraction of the sun and moon on the
prominent matter at the equator. In comparing the periods of the
revolutions of Jupiter and Saturn with the times of their rotation, it
appears that a year of Jupiter contains nearly ten thousand of his days,
and that of Saturn about thirty thousand Saturnian days.
The appearance of Saturn is unparalleled in the system of the world; he
is surrounded by a ring even brighter than himself, which always remains
in the plane of his equator, and viewed with a very good telescope, it
is found to consist of two concentric rings, divided by a dark band. By
the laws of mechanics, it is impossible that this body can retain its
position by the adhesion of its particles alone; it must necessarily
revolve with a velocity that will generate a centrifugal force
sufficient to balance the attraction of Saturn. Observation confirms the
truth of these principles, showing that the rings rotate about the
planet in 10 1/2 hours, which is considerably less than the time a
satellite would take to revolve about Saturn at the same distance. Their
plane is inclined to the ecliptic at an angle of 31°; and in consequence
of this obliquity of position they always appear elliptical to us, but
with an eccentricity so variable as even to be occasionally like a
straight line drawn across the planet. At present the apparent axes of
the rings are as 1000 to 160; and on the 29th of September, 1832,
the plane of the rings will pass through the centre of the earth
when they will be visible only with superior instruments, and will
appear like a fine line across the disc of Saturn. On the 1st of
December in the same year, the plane of the rings will pass through
the centre of the sun.
It is a singular result of the theory, that the rings could not maintain
their stability of rotation if they were everywhere of uniform
thickness; for the smallest disturbance would destroy the equilibrium,
which would become more and more deranged, till at last they would be
precipitated on the surface of the planet. The rings of Saturn must
therefore be irregular solids of unequal breadth in the different parts
of the circumference, so that their centres of gravity do not coincide
with the centres of their figures.
Professor Struve has also discovered that the centre of the ring is not
concentric with the centre of Saturn; the interval between the outer
edge of the globe of the planet and the outer edge of the ring on one
side, is 11"·073, and on the other side the interval is 11"·288;
consequently there is an eccentricity of the globe in the ring of
0"·215.
If the rings obeyed different forces, they would not remain in the same
plane, but the powerful attraction of Saturn always maintains them and
his satellites in the plane of his equator. The rings, by their mutual
action, and that of the sun and satellites, must oscillate about the
centre of Saturn, and produce phenomena of light and shadow, whose
periods extend to many years.
The periods of the rotation of the moon and the other satellites are
equal to the times of their revolutions, consequently these bodies
always turn the same face to their primaries; however, as the mean
motion of the moon is subject to a secular inequality which will
ultimately amount to many circumferences, if the rotation of the moon
were perfectly uniform, and not affected by the same inequalities, it
would cease exactly to counterbalance the motion of revolution; and the
moon, in the course of ages, would successively and gradually discover
every point other surface to the earth. But theory proves that this
never can happen; for the rotation of the moon, though it does not
partake of the periodic inequalities of her revolution, is affected by
the same secular variations, so that her motions of rotation and
revolution round the earth will always balance each other, and remain
equal. This circumstance arises from the form of the lunar spheroid,
which has three principal axes of different lengths at right angles to
each other. The moon is flattened at the poles from her centrifugal
force, therefore her polar axis is least; the other two are in the plane
of her equator, but that directed towards the earth is the greatest. The
attraction of the earth, as if it had drawn out that part of the moon's
equator, constantly brings the greatest axis, and consequently the same
hemisphere towards us, which makes her rotation participate in the
secular variations in her mean motion of revolution. Even if the angular
velocities of rotation and revolution had not been nicely balanced in
the beginning of the moon's motion, the attraction of the earth would
have recalled the greatest axis to the direction of the line joining the
centres of the earth and moon; so that it would vibrate on each side of
that line in the same manner as a pendulum oscillates on each side of
the vertical from the influence of gravitation.
No such libration is perceptible; and as the smallest disturbance would
make it evident, it is clear that if the moon has ever been touched by a
comet, the mass of the latter must have been extremely small; for if it
had been only the hundred-thousandth part of that of the earthy it would
have rendered the libration sensible. A similar libration exists in the
motions of Jupiter's satellites; but although the comet of 1767 and 1779
passed through the midst of them, their libration still remains
insensible. It is true, the moon is liable to librations depending on
the position of the spectator; at her rising, part of the western edge
of her disc is visible, which is invisible at her setting, and the
contrary takes place with regard to her eastern edge. There are also
librations arising from the relative positions of the earth and moon in
their respective orbits, but as they are only optical appearances, one
hemisphere will be eternally concealed from the earth. For the same
reason, the earth, which must be so splendid an object to one lunar
hemisphere, will be for ever veiled from the other. On account of these
circumstances, the remoter hemisphere of the moon has its day a
fortnight long, and a night of the same duration not even enlightened by
a moon, while the favoured side is illuminated by the reflection of the
earth during its long night. A moon exhibiting a surface thirteen times
larger than ours, with all the varieties of clouds, land, and water
coming successively into view, would be a splendid object to a lunar
traveller in a journey to his antipodes.
The great height of the lunar mountains probably has a considerable
influence on the phenomena of her motion, the more so as her compression
is small, and her mass considerable.