We will now proceed to give the method of determining the latitude of
the axis of the vortex, at the time of its passage over any given
meridian, and at any given time. And afterwards we will give a brief
abstract from the record of the weather, for one sidereal period of the
moon, in order to compare the theory with observation.
[Illustration: Fig. 4]
In the above figure, the circle PER represents the earth, E the equator,
PP' the poles, T the centre of the earth, C the mechanical centre of the
terral vortex, M the moon, XX' the axis of the vortex, and A the point
where the radius vector of the moon pierces the surface of the earth. If
we consider the axis of the vortex to be the axis of equilibrium in the
system, it is evident that TC will be to CM, as the mass of the moon to
the mass of the earth. Now, if we take these masses respectively as 1 to
72.3, and the moon's mean distance at 238,650 miles, the mean value of
TC is equal to this number, divided by the sum of these masses,--_i.e._
the mean radius vector of the little orbit, described by the earth's
centre around the centre of gravity of the earth and moon, is equal
238650/(72.3+1) = 3,256 miles; and at any other distance of the moon, is
equal to that distance, divided by the same sum. Therefore, by taking CT
in the inverse ratio of the mean semi-diameter of the moon to the true
semi-diameter, we shall have the value of CT at that time. But TA is to
TC as radius to the cosine of the arc AR, and RR' are the points on the
earth's surface pierced by the axis of the vortex, supposing this axis
coincident with the pole of the lunar orbit. If this were so, the
calculation would be very short and simple; and it will, perhaps,
facilitate the investigation, by considering, for the present, that the
two axes do coincide.
In order, also, to simplify the question, we will consider the earth a
perfect sphere, having a diameter of 7,900 miles, equal to the actual
polar diameter, and therefore TA is equal to 3,950 miles.
In the spherical triangle given on next page, we have given the point A,
being the position of the moon in right ascension and declination in the
heavens, and considered as terrestrial latitude and longitude.
Therefore, PA is equal to the complement of the moon's declination, P
being the pole of the earth, and L being the pole of the lunar orbit; PL
is equal to the obliquity of the lunar orbit, with respect to the earth,
and is therefore given by finding the true inclination of the lunar
orbit at the time, equal EL, (E being the pole of the ecliptic,) also
the true longitude of the ascending node, and the obliquity of the
ecliptic PE. Now, as we are supposing the axis of the vortex parallel
to the pole of the lunar orbit, and to pierce the earth's surface at R,
ARL will evidently all be in the same plane; and, as in the case of A
and L, this plane passes through the earth's centre, ARL must all lie in
the same great circle. Having, therefore, the right ascension of A, and
the right ascension of L, we have the angle P. This gives us two sides,
and the included angle, to find the side LA. But we have before found
the arc AR; we therefore know LR. But in finding LA, we found both the
angles L and A, and therefore can find PR, which is equal to the
complement of the latitude sought.
[Illustration: Fig. 5]
We have thus indicated briefly the simple process by which we could find
the latitude of the axis of the central vortex, supposing it to be
always coincident with the pole of the lunar orbit. The true problem is
more complicated, and the principal modifications, indicated by the
theory, are abundantly confirmed by observation. The determination of
the inclination of the axis of the vortex, its position in space at a
given time, and the law of its motion, was a work of cheerless labor for
a long time. He that has been tantalized by hope for years, and ever on
the eve of realization, has found the vision vanish, can understand the
feeling which proceeds from frequent disappointment in not finding that,
whose existence is almost demonstrated; and more especially when the
approximation differs but slightly from the actual phenomena.
The chief difficulty at the outset of these investigations, arose from
the conflicting authority of astronomers in relation to the mass of the
moon. We are too apt to confound the precision of the laws of nature,
with the perfection of human theories. Man observes the phenomena of the
heavens, and derives his means of predicting what will be, from what has
been. Hence the motions of the heavenly bodies are known to within a
trifling amount of the truth; but it does not follow that the true
explanation is always given by theory. If this were so, the mass of the
moon (for instance) ought to be the same, whether deduced from the
principle of gravitation or from some other source. This is not so.
Newton found it 1/40 of that of the earth. La Place, from a profound
theoretical discussion of the tides, gave it as 1/58.6, while from other
sources he found a necessity of diminishing it still more, to 1/68, and
finally as low as 1/75. Bailly, Herschel, and others, from the nutation
of the earth's axis, only found 1/80, and the Baron Lindenau deduced the
mass from the same phenomenon 1/88. In a very recent work by Mr. Hind,
he uses this last value in certain computations, and remarks, that we
shall not be very far wrong in considering it as 1/80 of the mass of the
earth. This shows the uncertainty of the matter in 1852. If astronomy is
so perfect as to determine the parallax of a fixed star, which is almost
always less than one second, why is it that the mass of the moon is not
more nearly approximated? Every two weeks the sun's longitude is
affected by the position of the moon, alternately increasing and
diminishing it, by a quantity depending solely upon the relative mass of
the earth and moon, and is a gross quantity compared to the parallax of
a star. So, also, the horizontal parallax--the most palpable of all
methods--taken by different observers at Berlin, and the Cape of Good
Hope, (a very respectable base line, one would suppose,) makes the mass
of the moon greater than its value derived from nutation; the first
giving about 1/70, the last about 1/74.2. Does not this declare that it
is unsafe to depend too absolutely on the strict wording of the
Newtonian law of gravitation. Happily our theory furnishes us with the
correct value of the moon's mass, written legibly on the surface of the
earth; and it comes out nearly what these two phenomena always gave it,
viz.: 1/72.3 of that of the earth. In another place we shall inquire
into the cause of the discrepancy as given by the nutation of the earth.