But it may be asked: If there be a modifying force in astronomy derived
from another source than that of gravitation, why is it that the
elements of the various members of the system derived solely from
gravitation should be so perfect? To this it may be answered, that
although astronomers have endeavored to derive every movement in the
heavens from that great principle, they have but partially succeeded.
Let us not surrender our right of examining Nature to the authority of a
great name, nor call any man master, either in moral or physical
science. It is well known that Kepler's law of the planetary distances
and periods, is a direct consequence of the Newtonian Law of
gravitation, and that the squares of the periodic times ought to be
proportional to the cubes of the mean distances. These times are given
accurately by the planets themselves, by the interval elapsing between
two consecutive passages of the node, and as in the case of the ancient
planets we have observations for more than two thousand years past,
these times are known to the fraction of the second. The determination
of the distances however, depends on the astronomer, and a tyro in the
science might suppose that these distances were actually measured; and
so they are roughly; but the astronomer does not depend on his
instruments, he trusts to _analogy_, and the mathematical perfection of
a law, which in the abstract is true; but which he does not know is
rigidly exact when applied to physical phenomena. From the immense
distance of the planets and the smallness of the earth, man is unable to
command a base line sufficiently long, to make the horizontal parallax a
sensible angle for the more distant planets; and there are difficulties
of no small magnitude to contend with, with those that are the nearest.
In the occasional transit of Venus across the sun, however, he is
presented with a means of measuring on an enlarged scale, from which the
distance of the sun is determined; and by _analogy_ the distance of all
the planets. Even the parallax of the sun itself is only correct, by
supposing that the square of the periodic time of Venus is in the same
proportion to the square of the periodic time of the earth as the cube
of her distance is to the cube of the earth's distance. Our next nearest
planet is Mars, and observations on this planet at its opposition to the
sun, invariably give a larger parallax for the sun--Venus giving 8.5776"
while Mars gives about 10". It is true that the first is obtained under
more favorable circumstances; but this does not prove the last to be
incorrect. It is well known that the British Nautical Almanac contains a
list of stars lying in the path of the planet Mars about opposition,
(for the very purpose of obtaining a correct parallax,) that minute
differences of declination may be detected by simultaneous observations
in places having great differences of latitude. Yet strange to say, the
result is discredited when not conformable to the parallax given by
Venus. If then, we cannot trust the parallax of Mars, _a fortiori_, how
can we trust the parallax of Jupiter, and say that his mean distance
exactly corresponds to his periodic time? Let us suppose, for instance,
that the radius vector of Jupiter fell short of that indicated by
analogy by 10,000 miles, we say that it would be extremely difficult,
nay, utterly impossible, to detect it by instrumental means. Let not
astronomers, therefore, be too sure that there is not a modifying cause,
independent of gravitation, which they will yet have to recognize. The
moon's distance is about one-fourth of a million of miles, and Neptune's
2854 millions, or in the ratio of 10,000 to 1; yet even the moon's
parallax is not trusted in determining her mass, how then shall we
determine the parallax of Neptune? It is therefore _possible_ that the
effective action of the sun is in some small degree different, on the
different planets, whether due to the action of the ether, to the
similarity or dissimilarity of material elements, to the temperature of
the different bodies, or to all combined, is a question yet to be
considered.
As another evidence of the necessity of modifying the strict wording of
the Newtonian law, it is found that the disturbing action of Jupiter on
different bodies, gives different values for the mass of Jupiter. The
mass deduced from Jupiter's action on his satellites, is different from
that derived from the perturbations of Saturn, and this last does not
correspond with that given by Juno: Vesta also gives a different mass
from the comet of Encke, and both vary from the preceding values.[41]
In the analytical investigation of planetary disturbances, the
disturbing force is usually divided into a radial and tangential force;
the first changing the law of gravitation, to which law the elliptic
form of the orbit is due. The radial disturbing force, therefore, being
directed to or from the centre, can have no influence over the first law
of Kepler, which teaches that the radius vector of each planet having
the sun as the centre, describes equal areas in equal times. If the
radial disturbing force be exterior to the disturbed body, it will
diminish the central force, and cause a progressive motion in the
aphelion point of the orbit. In the case of the moon this motion is very
rapid, the apogee making an entire revolution in 3232 days. Does this,
however, correspond with the law of gravitation? Sir Isaac Newton, in
calculating the effect of the sun's disturbing force on the motion of
the moon's apogee, candidly concludes thus: "Idoque apsis summa singulis
revolutionibus progrediendo conficit 1d 31' 28". Apsis lunae est duplo
velocior circiter." As there was a necessity for reconciling this
stubborn fact with the theory, his followers have made up the deficiency
by resorting to the tangential force, or, as Clairant proposed, by
continuing the approximations to terms of a higher order, or to the
square of the disturbing force.
Now, in a circular orbit, this tangential force will alternately
increase and diminish the velocity of the disturbed body, without
producing any permanent derangement, the same result would obtain in an
elliptical orbit, if the position of the major axis were stationary. In
the case of the moon, the apogee is caused to advance by the disturbing
power of the radial force, and, consequently, an exact compensation is
not effected: there remains a small excess of velocity which geometers
have considered equivalent to a doubling of the radial force, and have
thus obviated the difficulty. To those not imbued with the profound
penetration of the modern analyst, there must ever appear a little
inconsistency in this result. The major axis of a planet's orbit depends
solely on the velocity of the planet at a given distance from the sun,
and the tangential portion of the disturbance due to the sun, and
impressed upon the moon, must necessarily increase and diminish
alternately the velocity of the moon, and interfere with the equable
description of the areas. If, then, there be left outstanding a small
excess of velocity over and above the elliptical velocity of the moon,
at the end of each synodical revolution, in consequence of the motion
impressed on the moon's apogee by the radial force, the _legitimate_
effect would be a small enlargement of the lunar orbit every revolution
in a rapidly-increasing ratio, until the moon would at last be taken
entirely away. In the great inequality of Jupiter and Saturn, this
tangential force is not compensated at each revolution, in consequence
of continual changes in the configuration of the two planets at their
heliocentric conjunctions, with respect to the perihelion of their
orbits, and the near commensurability of their periods; and the effect
of the tangential force is, in this case, legitimately impressed on the
major axes of the orbits. But why (we may ask) should not this also be
expended on the motion of the aphelion as well as in the case of the
moon? Astronomy can make no distinctions between the orbit of a planet
and the orbit of a satellite. And, we might also ask, why the tangential
resistance to the comet of Encke should not also produce a retrograde
motion in the apsides of the orbit, instead of diminishing its period?
To the honor of Newton, be it remembered, that he never resorted to an
explanation of this phenomenon, which would vitiate that fundamental
proposition of his theory, in which the major axis of the orbit is shown
to depend on the velocity at any given distance from the focus.
Some cause, however, exists to double the motion of the apogee, and
that there is an outstanding excess of orbital velocity due to the
tangential force, is also true. This excess may tell in the way
proposed, provided some other arrangement exists to _prevent_ a
permanent dilation of the lunar orbit; and this provision may be found
in the increasing density of the ether, which prevents the moon
overstepping the bounds prescribed by her own density, and the force of
the radial stream of the terral vortex. In the case of Jupiter and
Saturn, their mutual action is much less interfered with by change of
density in the ether in the enlarged or contracted orbit, and,
consequently, the effect is natural. Thus, we have in the law of density
of the ethereal medium a better safeguard to the stability of the
dynamical balance of the system, than in the profound and beautiful
Theorems of La Grange. It will, of course, occur to every one, that we
are not to look for the same law in every vortex, and it will,
therefore, appear as if the satellites of Jupiter, whose theory is so
well known, should render apparent any deviation between their periodic
times and the periodic times of the contiguous parts of the vortex,
which would obtain, if the density of the ether in the Jovian vortex
were not as the square roots of the distances directly. But, we have
shown how there can be a balance preserved, if the tangential resistance
of the vortex shall be equal and contrary at the different distances at
which the satellites are placed; that is, if these two forces shall
follow the same law. These are matters, however, for future
investigation.