A
PRELIMINARY DISSERTATION
ON THE
MECHANISM OF THE HEAVENS.
BY
MRS. SOMMERVILLE
PHILADELPHIA:
CAREY & LEA
1832
In order to convey some idea of the object of this work, it may be
useful to offer a few preliminary observations on the nature of the
subject which it is intended to investigate, and of the means that have
already been adopted with so much success to bring within the reach of
our faculties, those truths which might seem to be placed so far beyond
them.
All the knowledge we possess of external objects is founded upon
experience, which furnishes a knowledge of facts, and the comparison of
these facts establishes relations, from which, induction, the intuitive
belief that like causes will produce like effects, leads us to general
laws. Thus, experience teaches that bodies fall at the surface of the
earth with an accelerated velocity, and proportional to their masses.
Newton proved, by comparison, that the force which occasions the fall of
bodies at the earth's surface, is identical with that which retains the
moon in her orbit; and induction led him to conclude that as the moon is
kept in her orbit by the attraction of the earth, so the planets might
be retained in their orbits by the attraction of the sun. By such steps
he was led to the discovery of one of those powers with which the
Creator has ordained that matter should reciprocally act upon matter.
Physical astronomy is the science which compares and identifies the laws
of motion observed on earth with the motions that take place in the
heavens, and which traces, by an uninterrupted chain of deduction from
the great principle that governs the universe, the revolutions and
rotations of the planets, and the oscillations of the fluids at their
surfaces, and which estimates the changes the system has hitherto
undergone or may hereafter experience, changes which require millions of
years for their accomplishment.
The combined efforts of astronomers, from the earliest dawn of
civilization, have been requisite to establish the mechanical theory of
astronomy: the courses of the planets have been observed for ages with a
degree of perseverance that is astonishing, if we consider the
imperfection, and even the want of instruments. The real motions of the
earth have been separated from the apparent motions of the planets; the
laws of the planetary revolutions have been discovered; and the
discovery of these laws has led to the knowledge of the gravitation of
matter. On the other hand, descending from the principle of gravitation,
every motion in the system of the world has been so completely
explained, that no astronomical phenomenon can now be transmitted to
posterity of which the laws have not been determined.
Science, regarded as the pursuit of truth, which can only be attained by
patient and unprejudiced investigation, wherein nothing is too great to
be attempted, nothing so minute as to be justly disregarded, must ever
afford occupation of consummate interest and of elevated meditation. The
contemplation of the works of creation elevates the mind to the
admiration of whatever is great and noble, accomplishing the object of
all study, which in the elegant language of Sir James Mackintosh is to
inspire the love of truth, of wisdom, of beauty, especially of goodness,
the highest beauty, and of that supreme and eternal mind, which contains
all truth and wisdom, all beauty and goodness. By the love or delightful
contemplation and pursuit of these transcendent aims for their own sake
only, the mind of man is raised from low and perishable objects, and
prepared for those high destinies which are appointed for all those who
are capable of them.
The heavens afford the most sublime subject of study which can be
derived from science: the magnitude and splendour of the objects, the
inconceivable rapidity with which they move, and the enormous distances
between them, impress the mind with some notion of the energy that
maintains them in their motions with a durability to which we can see no
limits. Equally conspicuous is the goodness of the great First Cause in
having endowed man with faculties by which he can not only appreciate
the magnificence of his works, but trace, with precision, the operation
of his laws, use the globe he inhabits us a base wherewith to measure
the magnitude and distance of the sun and planets, and make the diameter
of the earth's orbit the first step of a scale by which he may ascend to
the starry firmament. Such pursuits, while they ennoble the mind, at the
same time inculcate humility, by showing that there is a barrier, which
no energy, mental or physical, can ever enable us to pass: that however
profoundly we may penetrate the depths of space, there still remain
innumerable systems compared with which those which seem so mighty to us
must dwindle into insignificance, or even become invisible; and that not
only man, but the globe he inhabits, nay the whole system of which it
forms so small a part, might be annihilated, and its extinction be
unperceived in the immensity or creation.
A complete acquaintance with Physical Astronomy can only be attained by
those who are well versed in the higher branches of mathematical and
mechanical science: such alone can appreciate the extreme beauty of
the results, and of the means by which these results are obtained.
Nevertheless a sufficient skill in analysis to follow the general
outline, to see the mutual dependence of the different parts of the
system, and to comprehend by what means some of the most extraordinary
conclusions have been arrived at, is within the reach of many who shrink
from the task, appalled by difficulties, which perhaps are not more
formidable than those incident to the study of the elements of every
branch of knowledge, and possibly overrating them by not making a
sufficient distinction between the degree of mathematical acquirement
necessary for making discoveries, and that which is requisite for
understanding what others have done. That the study of mathematics and
their application to astronomy are full of interest will be allowed by
all who have devoted their time and attention to these pursuits, and
they only can estimate the delight of arriving at truth, whether it be
in the discovery of a world, or of a new property of numbers.
It has been proved by Newton that a particle of matter placed without
the surface of a hollow sphere is attracted by it in the name manner as
if its mass, or the whole matter it contains, were collected in its
centre. The same is therefore true of a solid sphere which may be
supposed to consist of an infinite number of concentric hollow spheres.
This however is not the case with a spheroid, but the celestial bodies
are so nearly spherical, and at such remote distances from each other,
that they attract and are attracted as if each were a dense point
situate in its centre of gravity, a circumstance which greatly
facilitates the investigation of their motions.
The attraction of the earth on bodies at its surface in that latitude,
the square of whose sine is ⅓, is the same as if it were a sphere; and
experience shows that bodies there fall through 16.0697 feet in a
second. The mean distance of the moon from the earth is about sixty
times the mean radius of the earth. When the number 16.0697 is
diminished in the ratio of 1 to 3600, which is the square of the moon's
distance from the earth, it is found to be exactly the space the moon
would fall through in the first second of her descent to the earth, were
she not prevented by her centrifugal force, arising from the velocity
with which she moves in her orbit. So that the moon is retained in her
orbit by a force having the same origin and regulated by the same law
with that which causes a stone to fall at the earth's surface. The earth
may therefore be regarded as the centre of a force which extends to the
moon; but as experience shows that the action and reaction of matter are
equal and contrary, the moon must attract the earth with an equal and
contrary force.
Newton proved that a body projected in space will move in a conic
section, if it be attracted by a force directed towards a fixed point,
and having an intensity inversely as the square of the distance; but
that any deviation from that law will cause it to move in a curve of a
different nature. Kepler ascertained by direct observation that the
planets describe ellipses round the sun, and later observations show
that comets also move in conic sections: it consequently follows that
the sun attracts all the planets and comets inversely as the square of
their distances from his centre; the sun therefore is the centre of a
force extending indefinitely in space, and including all the bodies of
the system in its action.
Kepler also deduced from observation, that the squares of the periodic
times of the planets, or the times of their revolutions round the sun,
are proportional to the cubes of their mean distances from his centre:
whence it follows, that the intensity of gravitation of all the bodies
towards the sun is the same at equal distances; consequently gravitation
is proportional to the masses, for if the planets and comets be supposed
to be at equal distances from the sun and left to the effects of
gravity, they would arrive at his surface at the same time. The
satellites also gravitate to their primaries according to the same law
that their primaries do to the sun. Hence, by the law of action and
reaction, each body is itself the centre of an attractive force
extending indefinitely in space, whence proceed all the mutual
disturbances that render the celestial motions so complicated, and their
investigation so difficult.
The gravitation of matter directed to a centre, and attracting directly
as the mass, and inversely as the square of the distance, does not
belong to it when taken in mass; particle acts on particle according to
the same law when at sensible distances from each other. If the sun
acted on the centre of the earth without attracting each of its
particles, the tides would be very much greater than they now are, and
in other respects they also would be very different. The gravitation of
the earth to the sun results from the gravitation of all its particles,
which in their turn attract the sun in the ratio of their respective
masses. There is a reciprocal action likewise between the earth and
every particle at its surface; were this not the case, and were any
portion of the earth, however small, to attract another portion and not
be itself attracted, the centre of gravity of the earth would be moved
in space, which is impossible.
The form of the planets results from the reciprocal attraction of their
component particles. A detached fluid mass, if at rest, would assume the
form of a sphere, from the reciprocal attraction of its particles; but
if the mass revolves about an axis, it becomes flattened at the poles,
and bulges at the equator, in consequence of the centrifugal force
arising from the velocity of rotation. For, the centrifugal force
diminishes the gravity of the particles at the equator, and equilibrium
can only exist when these two forces are balanced by an increase of
gravity; therefore, as the attractive force is the same on all particles
at equal distances from the centre of a sphere, the equatorial particles
would recede from the centre till their increase in number balanced the
centrifugal force by their attraction, consequently the sphere would
become an oblate spheroid; and a fluid partially or entirely covering a
solid, as the ocean and atmosphere cover the earth, must assume that
form in order to remain in equilibrio. The surface of the sea is
therefore spheroidal, and the surface of the earth only deviates from
that figure where it rises above or sinks below the level of the sea;
but the deviation is so small that it is unimportant when compared with
the magnitude of the earth. Such is the form of the earth and planets,
but the compression or flattening at their poles is so small, that even
Jupiter, whose rotation is the most rapid, differs but little from a
sphere. Although the planets attract each other as if they were spheres
on account of their immense distances, yet the satellites are near
enough to be sensibly affected in their motions by the forms of their
primaries. The moon for example is so near the earth, that the
reciprocal attraction between each of her particles and each of the
particles in the prominent mass at the terrestrial equator, occasions
considerable disturbances in the motions of both bodies. For, the action
of the moon on the matter at the earth's equator produces a nutation in
the axis of rotation, and the reaction of that matter on the moon is the
cause of a corresponding nutation in the lunar orbit.
If a sphere at rest in space receives an impulse passing through its
centre of gravity, all its parts will move with an equal velocity in a
straight line; but if the impulse does not pass through the centre of
gravity, its particles having unequal velocities, will give it a
rotatory motion at the same time that it is translated in space. These
motions are independent of one another, so that a contrary impulse
passing through its centre of gravity will impede its progression,
without interfering with its rotation. As the sun rotates about an axis,
it seems probable if an impulse in a contrary direction has not been
given to his centre of gravity, that he moves in space accompanied by
all those bodies which compose the solar system, a circumstance that
would in no way interfere with their relative motions; for, in
consequence of our experience that force is proportional to velocity,
the reciprocal attractions of a system remain the same, whether its
centre of gravity be at rest, or moving uniformly in space. It is
computed that had the earth received its motion from a single impulse,
such impulse must have passed through a point about twenty-five miles
from its centre.
Since the motions of the rotation and translation of the planets are
independent of each other, though probably communicated by the same
impulse, they form separate subjects of investigation.
A planet moves in its elliptical orbit with a velocity varying every
instant, in consequence of two forces, one tending to the centre of the
sun, and the other in the direction of a tangent to its orbit, arising
from the primitive impulse given at the time when it was launched into
space: should the force in the tangent cease, the planet would fall to
the sun by its gravity; were the sun not to attract it, the planet would
fly off in the tangent. Thus, when a planet is in its aphelion or at the
point where the orbit is farthest from the sun, his action overcomes its
velocity, and brings it towards him with such an accelerated motion,
that it at last overcomes the sun's attraction, and shoots past him;
then, gradually decreasing in velocity, it arrives at the aphelion where
the sun's attraction again prevails. In this motion the radii vectores,
or imaginary lines joining the centres of the sun and planets, pass over
equal areas in equal times.
If the planets were attracted by the sun only, this would ever be their
course; and because his action is proportional to his mass, which is
immensely larger than that of all the planets put together, the
elliptical is the nearest approximation to their true motions, which are
extremely complicated, in consequence of their mutual attraction, so
that they do not move in any known or symmetrical curve, but in paths
now approaching to, and now receding from the elliptical form, and their
radii vectores do not describe areas exactly proportional to the time.
Thus the areas become a test of the existence of disturbing forces.
To determine the motion of each body when disturbed by all the rest is
beyond the power of analysis; it is therefore necessary to estimate the
disturbing action of one planet at a time, whence arises the celebrated
problem of the three bodies, which originally was that of the moon, the
earth, and the sun, namely,--the masses being given of three bodies
projected from three given points, with velocities given both in
quantity and direction; and supposing the bodies to gravitate to one
another with forces that are directly as their masses, and inversely as
the squares of the distances, to find the lines described by these
bodies, and their position at any given instant.
By this problem the motions of translation of all the celestial bodies
are determined. It is one of extreme difficulty, and would be of
infinitely greater difficulty, if the disturbing action were not very
small, when compared with the central force. As the disturbing influence
of each body may be found separately, it is assumed that the action of
the whole system in disturbing any one planet is equal to the sum of all
the particular disturbances it experiences, on the general mechanical
principle, that the sum of any number of small oscillations is nearly
equal to their simultaneous and joint effect.
On account of the reciprocal action of matter, the stability of the
system depends on the intensity of the primitive momentum of the
planets, and the ratio of their masses to that of the sun: for the
nature of the conic sections in which the celestial bodies move, depends
on the velocity with which they were first propelled in space; had that
velocity been such as to make the planets move in orbits of unstable
equilibrium, their mutual attractions might have changed them into
parabolas or even hyperbolas; so that the earth and planets might ages
ago have been sweeping through the abyss of space: but as the orbits
differ very little from circles, the momentum of the planets when
projected, must have been exactly sufficient to ensure the permanency
and stability of the system. Besides the mass of the sun is immensely
greater than those of the planets; and as their inequalities bear the
same ratio to their elliptical motions as their masses do to that of the
sun, their mutual disturbances only increase or diminish the
eccentricities of their orbits by very minute quantities; consequently
the magnitude of the sun's mass is the principal cause of the stability
of the system. There is not in the physical world a more splendid
example of the adaptation of means to the accomplishment of the end,
than is exhibited in the nice adjustment of these forces.
The orbits of the planets have a very small inclination to the plane of
the ecliptic in which the earth moves; and on that account, astronomers
refer their motions to it at a given epoch as a known and fixed
position. The paths of the planets, when their mutual disturbances are
omitted, are ellipses nearly approaching to circles, whose planes,
slightly inclined to the ecliptic: cut it in straight lines passing
through the centre of the sun; the points where the orbit intersects the
plane of the ecliptic are its nodes.
The orbits of the recently discovered planets deviate more from the
ecliptic: that of Pallas has an inclination of 35° to it: on that
account it will be more difficult to determine their motions. These
little planets have no sensible effect in disturbing the rest, though
their own motions are rendered very irregular by the proximity of
Jupiter and Saturn.
The planets are subject to disturbances of two distinct kinds, both
resulting from the constant operation of their reciprocal attraction,
one kind depending upon their positions with regard to each other,
begins from zero, increases to a maximum, decreases and becomes zero
again, when the planets return to the same relative positions. In
consequence of these, the troubled planet is sometimes drawn away from
the sun, sometimes brought nearer to him; at one time it is drawn above
the plane of its orbit, at another time below it, according to the
position of the disturbing body. All such changes, being accomplished in
short periods, some in a few months, others in years, or in hundreds of
years, are denominated Periodic Inequalities.
The inequalities of the other kind, though occasioned likewise by the
disturbing energy of the planets, are entirely independent of their
relative positions; they depend on the relative positions of the orbits
alone, whose forms and places in space are altered by very minute
quantities in immense periods of time, and are therefore called Secular
Inequalities.
In consequence of disturbances of this kind, the apsides, or extremities
of the major axes of all the orbits, have a direct, but variable motion
in space, excepting those of Venus, which are retrograde; and the lines
of the nodes move with a variable velocity in the contrary direction.
The motions of both are extremely slow; it requires more than 109770
years for the major axis of the earth's orbit to accomplish a sidereal
revolution, and 20935 years to complete its tropical motion. The major
axis of Jupiter's orbit requires no less than 197561 years to perform
its revolution from the disturbing action of Saturn alone. The periods
in which the nodes revolve are also very great. Beside these, the
inclination and eccentricity of every orbit are in a state of perpetual,
but slow change. At the present time, the inclinations of all the orbits
are decreasing; but so slowly, that the inclination of Jupiter's orbit
is only six minutes less now than it was in the age of Ptolemy. The
terrestrial eccentricity is decreasing at the rate of 3914 miles in a
century; and if it were to decrease equably, it would be 36300 years
before the earth's orbit became a circle. But in the midst of all these
vicissitudes, the major axes and mean motions of the planets remain
permanently independent of secular changes; they are so connected by
Kepler's law of the squares of the periodic times being proportional to
the cubes of the mean distances of the planets from the sun, that one
cannot vary without affecting the other.
With the exception of these two elements, it appears, that all the
bodies are in motion, and every orbit is in a state of perpetual change.
Minute as these changes are, they might be supposed liable to accumulate
in the course of ages sufficiently to derange the whole order of nature,
to alter the relative positions of the planets, to put an end to the
vicissitudes of the seasons, and to bring about collisions, which would
involve our whole system, now so harmonious, in chaotic confusion. The
consequences being so dreadful, it is natural to inquire, what proof
exists that creation will be preserved from such a catastrophe? For
nothing can be known from observation, since the existence of the human
race has occupied but a point in duration, while these vicissitudes
embrace myriads of ages. The proof is simple and convincing. All the
variations of the solar system, as well secular as periodic, are
expressed analytically by the sines and cosines of circular arcs, which
increase with the time; and as a sine or cosine never can exceed the
radius, but must oscillate between zero and unity, however much the time
may increase, it follows, that when the variations have by slow changes
accumulated in however long a time to a maximum, they decrease by the
same slow degrees, till they arrive at their smallest value, and then
begin a new course, thus for ever oscillating about a mean value. This,
however, would not be the case if the planets moved in a resisting
medium, for then both the eccentricity and the major axes of the orbits
would vary with the time, so that the stability of the system would be
ultimately destroyed. But if the planets do move in an ethereal medium,
it must be of extreme rarity, since its resistance has hitherto been
quite insensible.
Three circumstances have generally been supposed necessary to prove the
stability of the system: the small eccentricities of the planetary
orbits, their small inclinations, and the revolution of all the bodies,
as well planets as satellites, in the same direction. These, however,
are not necessary conditions: the periodicity of the terms in which the
inequalities are expressed is sufficient to assure us, that though we do
not know the extent of the limits, nor the period of that grand cycle
which probably embraces millions of years, yet they never will exceed
what is requisite for the stability and harmony of the whole, for the
preservation of which every circumstance is so beautifully and
wonderfully adapted.
The plane of the ecliptic itself, though assumed to be fixed at a given
epoch for the convenience of astronomical computation, is subject to a
minute secular variation of 52"·109, occasioned by the reciprocal action
of the planets; but as this is also periodical, the terrestrial equator,
which is inclined to it at an angle of about 23° 28', will never
coincide with the plane of the ecliptic; so there never can be perpetual
spring.
The rotation of the earth is uniform; therefore day and night, summer
and winter, will continue their vicissitudes while the system endures,
or is untroubled by foreign causes.
Yonder starry sphere
Of planets, and of fix'd, in all her wheels
Resembles nearest, mazes intricate,
Eccentric, intervolv'd, yet regular
Then most, when most irregular they seem.
The stability of our system was established by La Grange, 'a discovery,'
says Professor Playfair, 'that must render the name for ever memorable
in science, and revered by those who delight in the contemplation of
whatever is excellent and sublime. After Newton's discovery of the
elliptical orbits of the planets, La Grange's discovery of their
periodical inequalities is without doubt the noblest truth in physical
astronomy; and, in respect of the doctrine of final causes, it may be
regarded as the greatest of all.'
Notwithstanding the permanency of our system, the secular variations in
the planetary orbits would have been extremely embarrassing to
astronomers, when it became necessary to compare observations separated
by long periods. This difficulty is obviated by La Place, who has shown
that whatever changes time may induce either in the orbits themselves,
or in the plane of the ecliptic, there exists an invariable plane
passing through the centre of gravity of the sun, about which the whole
system oscillates within narrow limits, and which is determined by this
property; that if every body in the system be projected on it, and if
the mass of each be multiplied by the area described in a given time by
its projection on this plane, the sum of all these products will be a
maximum. This plane of greatest inertia, by no means peculiar to the
solar system, but existing in every system of bodies submitted to their
mutual attractions only, always remains parallel to itself, and
maintains a fixed position, whence the oscillations of the system may be
estimated through unlimited time. It is situate nearly half way between
the orbits of Jupiter and Saturn, and is inclined to the ecliptic at an
angle of about 1° 35' 31".
All the periodic and secular inequalities deduced from the law of
gravitation are so perfectly confirmed by observations, that analysis
has become one of the most certain means of discovering the planetary
irregularities, either when they are too small, or too long in their
periods, to be detected by other methods. Jupiter and Saturn, however,
exhibit inequalities which for a long time seemed discordant with that
law. All observations, from those of the Chinese and Arabs down to the
present day, prove that for ages the mean motions of Jupiter and Saturn
have been affected by great inequalities of very long periods, forming
what appeared an anomaly in the theory of the planets. It was long known
by observation, that five times the mean motion of Saturn is nearly
equal to twice that of Jupiter; a relation which the sagacity of La
Place perceived to be the cause of a periodic inequality in the mean
motion of each of these planets, which completes its period in nearly
929 Julian years, the one being retarded, while the other is
accelerated. These inequalities are strictly periodical, since they
depend on the configuration of the two planets; and the theory is
perfectly confirmed by observation, which shows that in the course of
twenty centuries, Jupiter's mean motion has been accelerated by 3° 23',
and Saturn's retarded by 5° 13'.
It might be imagined that the reciprocal action of such planets as have
satellites would be different from the influence of those that have
none; but the distances of the satellites from their primaries are
incomparably less than the distances of the planets from the sun, and
from one another, so that the system of a planet and its satellites
moves nearly as if all those bodies were united in their common centre
of gravity; the action of the sun however disturbs in some degree the
motion of the satellites about their primary.
The changes that take place in the planetary system are exhibited on a
small scale by Jupiter and his satellites; and as the period requisite
for the development of the inequalities of these little moons only
extends to a few centuries, it may be regarded as an epitome of that
grand cycle which will not be accomplished by the planets in myriads of
centuries. The revolutions of the satellites about Jupiter are precisely
similar to those of the planets about the sun; it is true they are
disturbed by the sun, but his distance is so great, that their motions
are nearly the same as if they were not under his influence. The
satellites like the planets, were probably projected in elliptical
orbits, but the compression of Jupiter's spheroid is very great in
consequence of his rapid rotation; and as the masses of the satellites
are nearly 100000 times less than that of Jupiter, the immense quantity
of prominent matter at his equator must soon have given the circular
form observed in the orbits of the first and second satellites, which
its superior attraction will always maintain. The third and fourth
satellites being further removed from its influence, move in orbits with
a very small eccentricity. The same cause occasions the orbits of the
satellites to remain nearly in the plane of Jupiter's equator, on
account of which they are always seen nearly in the same line; and the
powerful action of that quantity of prominent matter is the reason why
the motion of the nodes of these little bodies is so much more rapid
than those of the planet. The nodes of the fourth satellite accomplish a
revolution in 520 years, while those of Jupiter's orbit require no less
than 50673 years, a proof of the reciprocal attraction between each
particle of Jupiter's equator and of the satellites. Although the two
first satellites sensibly move in circles, they acquire a small
ellipticity from the disturbances they experience.
The orbits of the satellites do not retain a permanent inclination,
either to the plane of Jupiter's equator, or to that of his orbit, but
to certain planes passing between the two, and through their
intersection; these have a greater inclination to his equator the
further the satellite is removed, a circumstance entirely owing to the
influence of Jupiter's compression.
A singular law obtains among the mean motions and mean longitudes of the
three first satellites. It appears from observation, that the mean
motion of the first satellite, plus twice that of the third, is equal to
three times that of the second, and that the mean longitude of the first
satellite, minus three times that of the second, plus twice that of the
third, is always equal to two right angles. It is proved by theory, that
if these relations had only been approximate when the satellites were
first launched into space, their mutual attractions would have
established and maintained them. They extend to the synodic motions of
the satellites, consequently they affect their eclipses, and have a very
great influence on their whole theory. The satellites move so nearly in
the plane of Jupiter's equator, which has a very small inclination to
his orbit, that they are frequently eclipsed by the planet. The instant
of the beginning or end of an eclipse of a satellite marks the same
instant of absolute time to all the inhabitants of the earth; therefore
the time of these eclipses observed by a traveller, when compared with
the time of the eclipse computed for Greenwich or any other fixed
meridian, gives the difference of the meridians in time, and
consequently the longitude of the place of observation. It has required
all the refinements of modern instruments to render the eclipses of
these remote moons available to the mariner; now however, that system of
bodies invisible to the naked eye, known to man by the aid of science
alone, enables him to traverse the ocean, spreading the light of
knowledge and the blessings of civilization over the most remote
regions, and to return loaded with the productions of another
hemisphere. Nor is this all: the eclipses of Jupiter's satellites have
been the means or a discovery, which, though not so immediately
applicable to the wants of man, unfolds a property of light, that
medium, without whose cheering influence all the beauties of the
creation would have been to us a blank. It is observed, that those
eclipses of the first satellite which happen when Jupiter is near
conjunction, are later by 16' 26" than those which take place when the
planet is in opposition. But as Jupiter is nearer to us when in
opposition by the whole breadth of the earth's orbit than when in
conjunction, this circumstance was attributed to the time employed by
the rays of light in crossing the earth's orbit, a distance of 192
millions of miles; whence it is estimated, that light travels at the
rate of 192000 miles in one second. Such is its velocity, that the
earth, moving at the rate of nineteen miles in a second, would take two
months to pass through a distance which a ray of light would dart over
in eight minutes. The subsequent discovery of the aberration of light
confirmed this astonishing result.
Objects appear to be situate in the direction of the rays that proceed
from them. Were light propagated instantaneously, every object, whether
at rest or in motion, would appear in the direction of these rays; but
as light takes some time to travel, when Jupiter is in conjunction, we
see him by means of rays that left him 16' 26" before; but during that
time we have changed our position, in consequence of the motion of the
earth in its orbit; we therefore refer Jupiter to a place in which he is
not. His true position is in the diagonal of the parallelogram, whose
sides are in the ratio of the velocity of light to the velocity of the
earth in its orbit, which is as 192000 to 19. In consequence of
aberration, none of the heavenly bodies are in the place in which they
seem to be. In fact, if the earth were at rest, rays from a star would
pass along the axis of a telescope directed to it; but if the earth were
to begin to move in its orbit with its usual velocity, these rays would
strike against the side of the tube; it would therefore be necessary to
incline the telescope a little, in order to see the star. The angle
contained between the axis of the telescope and a line drawn to the true
place of the star, is its aberration, which varies in quantity and
direction in different parts of the earth's orbit; but as it never
exceeds twenty seconds, in ordinary cases.
The velocity of light deduced from the observed aberration of the fixed
stars, perfectly corresponds with that given by the eclipses of the
first satellite. The same result obtained from sources so different,
leaves not a doubt of its truth. Many such beautiful coincidences,
derived from apparently the most unpromising and dissimilar
circumstances, occur in physical astronomy, and prove dependences which
we might otherwise be unable to trace. The identity of the velocity of
light at the distance of Jupiter and on the earth's surface shows that
its velocity is uniform; and if light consists in the vibrations of an
elastic fluid or ether filling space, which hypothesis accords best with
observed phenomena, the uniformity of its velocity shows that the
density of the fluid throughout the whole extent of the solar system,
must be proportional to its elasticity. Among the fortunate conjectures
which have been confirmed by subsequent experience, that of Bacon is not
the least remarkable. "It produces in me," says the restorer of true
philosophy, "a doubt, whether the face of the serene and starry heavens
be seen at the instant it really exists, or not till some time later;
and whether there be not, with respect to the heavenly bodies, a true
time and an apparent time, no less than a true place and an apparent
place, as astronomers say, on account of parallax. For it seems
incredible that the species or rays of the celestial bodies can pass
through the immense interval between them and us in an instant; or that
they do not even require some considerable portion of time."
As great discoveries generally lead to a variety of conclusions, the
aberration of light affords a direct proof of the motion of the earth in
its orbit; and its rotation is proved by the theory of falling bodies,
since the centrifugal force it induces retards the oscillations of the
pendulum in going from the pole to the equator. Thus a high degree of
scientific knowledge has been requisite to dispel the errors of the
senses.
The little that is known of the theories of the satellites of Saturn and
Uranus is in all respects similar to that of Jupiter. The great
compression of Saturn occasions its satellites to move nearly in the
plane of its equator. Of the situation of the equator of Uranus we know
nothing, nor of its compression. The orbits of its satellites are nearly
perpendicular to the plane of the ecliptic.
Our constant companion the moon next claims attention. Several
circumstances concur to render her motions the most interesting, and at
the same time the most difficult to investigate of all the bodies of our
system. In the solar system planet troubles planet, but in the lunar
theory the sun is the great disturbing cause; his vast distance being
compensated by his enormous magnitude, so that the motions of the moon
are more irregular than those of the planets; and on account of the
great ellipticity of her orbit and the size of the sun, the
approximations to her motions are tedious and difficult, beyond what
those unaccustomed to such investigations could imagine. Neither the
eccentricity of the lunar orbit, nor its inclination to the plane of the
ecliptic, have experienced any changes from secular inequalities; but
the mean motion, the nodes, and the perigee, are subject to very
remarkable variations.
From an eclipse observed at Babylon by the Chaldeans, on the 19th of
March, seven hundred and twenty-one years before the Christian era, the
place of the moon is known from that of the sun at the instant of
opposition; whence her mean longitude may be found; but the comparison
of this mean longitude with another mean longitude, computed back for
the instant of the eclipse from modern observations, shows that the moon
performs her revolution round the earth more rapidly and in a shorter
time now, than she did formerly; and that the acceleration in her mean
motion has been increasing from age to age as the square of the time;
all the ancient and intermediate eclipses confirm this result. As the
mean motions of the planets have no secular inequalities, this seemed to
be an unaccountable anomaly, and it was at one time attributed to the
resistance of an ethereal medium pervading space; at another to the
successive transmission of the gravitating force: but as La Place proved
that neither of these causes, even if they exist, have any influence on
the motions of the lunar perigee or nodes, they could not affect the
mean motion, a variation in the latter from such a cause being
inseparably connected with variations in the two former of these
elements. That great mathematician, however, in studying the theory of
Jupiter's satellites, perceived that the secular variations in the
elements of Jupiter's orbit, from the action of the planets, occasion
corresponding changes in the motions of the satellites: this led him to
suspect that the acceleration in the mean motion of the moon might be
connected with the secular variation in the eccentricity of the
terrestrial orbit; and analysis has proved that he assigned the true
cause.
If the eccentricity of the earth's orbit were invariable, the moon would
be exposed to a variable disturbance from the action of the sun, in
consequence of the earth's annual revolution; but it would be periodic,
since it would be the same as often as the sun, the earth, and the moon
returned to the same relative positions: on account however of the slow
and incessant diminution in the eccentricity of the terrestrial orbit,
the revolution of our planet is performed at different distances from
the sun every year. The position of the moon with regard to the sun,
undergoes a corresponding change; so that the mean action of the sun on
the moon varies from one century to another, and occasions the secular
increase in the moon's velocity called the acceleration, a name which is
very appropriate in the present age, and which will continue to be so
for a vast number of ages to come; because, as long as the earth's
eccentricity diminishes, the moon's mean motion will be accelerated; but
when the eccentricity has passed its minimum and begins to increase, the
mean motion will be retarded from age to age. At present the secular
acceleration is about 10", but its effect on the moon's place increases
as the square of the time. It is remarkable that the action of the
planets thus reflected by the sun to the moon, is much more sensible
than their direct action, either on the earth or moon. The secular
diminution in the eccentricity, which has not altered the equation of
the centre of the sun by eight minutes since the earliest recorded
eclipses, has produced a variation of 1° 48' in the moon's longitude,
and of 7° 12' in her mean anomaly.
The action of the sun occasions a rapid but variable motion in the nodes
and perigee of the lunar orbit; the former, though they recede during
the greater part of the moon's revolution, and advance during the
smaller, perform their sidereal revolutions in 6793^days.4212, and the
latter, though its motion is sometimes retrograde and sometimes direct,
in 3232^days.5807, or a little more than nine years: but such is the
difference between the disturbing energy of the sun and that of all the
planets put together, that it requires no less than 109770 years for the
greater axis of the terrestrial orbit to do the same. It is evident that
the same secular variation which changes the sun's distance from the
earth, and occasions the acceleration in the moon's mean motion, must
affect the motion of the nodes and perigee; and it consequently appears,
from theory as well as observation, that both these elements are subject
to a secular inequality, arising from the variation in the eccentricity
of the earth's orbit, which connects them with the acceleration; so that
both are retarded when the mean motion is anticipated. The secular
variations in these three elements are in the ratio of the numbers 3,
0.735, and 1; whence the three motions of the moon, with regard to the
sun, to her perigee, and to her nodes, are continually accelerated, and
their secular equations are as the numbers 1, 4, and 0.265, or according
to the most recent investigations as 1, 4, 6776 and 0.391. A comparison
of ancient eclipses observed by the Arabs, Greeks, and Chaldeans,
imperfect as they are, with modern observations, perfectly confirms
these results of analysis.
Future ages will develop these great inequalities, which at some most
distant period will amount to many circumferences. They are indeed
periodic; but who shall tell their period? Millions of years must elapse
before that great cycle is accomplished; but 'such changes, though rare
in time, are frequent in eternity.'
The moon is so near, that the excess of matter at the earth's equator
occasions periodic variations in her longitude and latitude; and, as the
cause must be proportional to the effect, a comparison of these
inequalities, computed from theory, with the same given by observation,
shows that the compression of the terrestrial spheroid, or the ratio of
the difference between the polar and equatorial diameter to the diameter
of the equator is 1/305.05. It is proved analytically, that if a fluid
mass of homogeneous matter, whose particles attract each other inversely
as the square of the distance, were to revolve about an axis, as the
earth, it would assume the form of a spheroid, whose compression is
1/230. Whence it appears, that the earth is not homogeneous, but decreases
in density from its centre to its circumference. Thus the moon's eclipses
show the earth to be round, and her inequalities not only determine the
form, but the internal structure of our planet; results of analysis which
could not have been anticipated. Similar inequalities in Jupiter's
satellites prove that his mass is not homogeneous, and that his
compression is 1/13·8.
The motions of the moon have now become of more importance to the
navigator and geographer than those of any other body, from the
precision with which the longitude is determined by the occultations of
stars and lunar distances. The lunar theory is brought to such
perfection, that the times of these phenomena, observed under any
meridian, when compared with that computed for Greenwich in the Nautical
Almanack, gives the longitude of the observer within a few miles. The
accuracy of that work is obviously of extreme importance to a maritime
nation; we have reason to hope that the new Ephemeris, now in
preparation, will be by far the most perfect work of the kind that ever
has been published.
From the lunar theory, the mean distance of the sun from the earth, and
thence the whole dimensions of the solar system are known; for the
forces which retain the earth and moon in their orbits, are respectively
proportional to the radii vectores of the earth and moon, each being
divided by the square of its periodic time; and as the lunar theory
gives the ratio of the forces, the ratio of the distance of the sun and
moon from the earth is obtained: whence it appears that the sun's
distance from the earth is nearly 396 times greater than that of the
moon.
The method however of finding the absolute distances of the celestial
bodies in miles, is in fact the same with that employed in measuring
distances of terrestrial objects. From the extremities of a known base
the angles which the visual rays from the object form with it, are
measured; their sum subtracted from two right-angles gives the angle
opposite the base; therefore by trigonometry, all the angles and sides
of the triangle may be computed; consequently the distance of the object
is found. The angle under which the base of the triangle is seen from
the object, is the parallax of that object; it evidently increases and
decreases with the distance; therefore the base must be very great
indeed, to be visible at all from the celestial bodies. But the globe
itself whose dimensions are ascertained by actual admeasurement,
furnishes a standard of measures, with which we compare the distances,
masses, densities, and volumes of the sun and planets.
The courses of the great rivers, which are in general navigable to a
considerable extent, prove that the curvature of the land differs but
little from that of the ocean; and as the heights of the mountains and
continents are, at any rate, quite inconsiderable when compared with the
magnitude of the earth, its figure is understood to be determined by a
surface at every point perpendicular to the direction of gravity, or of
the plumb-line, and is the same which the sea would have if it were
continued all round the earth beneath the continents. Such is the figure
that has been measured in the following manner:--