e earth were sufficiently sensible not to be confounded with the
errors of observation. It was accordingly necessary to find the general
formula of perturbations of this nature, in order to be able, as in the
case of the solar parallax, to eliminate the unknown quantity.

The ardour of Laplace, combined with his power of analytical research,
surmounted all obstacles. By means of an investigation which demanded
the most minute attention, the great geometer discovered in the theory
of the moon's movements, two well-defined perturbations depending on the
spheroidal figure of the earth. The first affected the resolved element
of the motion of our satellite which is chiefly measured with the
instrument known in observatories by the name of the transit instrument;
the second, which operated in the direction north and south, could only
be effected by observations with a second instrument termed the mural
circle. These two inequalities of very different magnitudes connected
with the cause which produces them by analytical combinations of totally
different kinds have, however, both conducted to the same value of the
ellipticity. It must be borne in mind, however, that the ellipticity
thus deduced from the movements of the moon, is not the ellipticity
corresponding to such or such a country, the ellipticity observed in
France, in England, in Italy, in Lapland, in North America, in India, or
in the region of the Cape of Good Hope, for the earth's materials having
undergone considerable upheavings at different times and in different
places, the primitive regularity of its curvature has been sensibly
disturbed by this cause. The moon, and it is this circumstance which
renders the result of such inestimable value, ought to assign, and has
in reality assigned the general ellipticity of the earth; in other
words, it has indicated a sort of mean value of the various
determinations obtained at enormous expense, and with infinite labour,
as the result of long voyages undertaken by astronomers of all the
countries of Europe.

I shall add a few brief remarks, for which I am mainly indebted to the
author of the _Mecanique Celeste_. They seem to be eminently adapted for
illustrating the profound, the unexpected, and almost paradoxical
character of the methods which I have just attempted to sketch.

What are the elements which it has been found necessary to confront with
each other in order to arrive at results expressed even to the precision