he 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. W