eneous.[29] When in attempting the solution of great problems
we have recourse to such simplifications; when, in order to elude
difficulties of calculation, we depart so widely from natural and
physical conditions, the results relate to an ideal world, they are in
reality nothing more than flights of the imagination.

In order to apply mathematical analysis usefully to the determination of
the figure of the earth it was necessary to abandon all idea of
homogeneity, all constrained resemblance between the forms of the
superposed and unequally dense strata; it was necessary also to examine
the case of a central solid nucleus. This generality increased tenfold
the difficulties of the problem; neither Clairaut nor D'Alembert was,
however, arrested by them. Thanks to the efforts of these two eminent
geometers, thanks to some essential developments due to their immediate
successors, and especially to the illustrious Legendre, the theoretical
determination of the figure of the earth has attained all desirable
perfection. There now reigns the most satisfactory accordance between
the results of calculation and those of direct measurement. The earth,
then, was originally fluid: analysis has enabled us to ascend to the
earliest ages of our planet.[30]

In the time of Alexander comets were supposed by the majority of the
Greek philosophers to be merely meteors generated in our atmosphere.
During the middle ages, persons, without giving themselves much concern
about the nature of those bodies, supposed them to prognosticate
sinister events. Regiomontanus and Tycho Brahe proved by their
observations that they are situate beyond the moon; Hevelius, Doerfel,
&c., made them revolve around the sun; Newton established that they move
under the immediate influence of the attractive force of that body, that
they do not describe right lines, that, in fact, they obey the laws of
Kepler. It was necessary, then, to prove that the orbits of comets are
curves which return into themselves, or that the same comet has been
seen on several distinct occasions. This discovery was reserved for
Halley. By a minute investigation of the circumstances connected with
the apparitions of all the comets to be met with in the records of
history, in ancient chronicles, and in astronomical annals, this eminent
philosopher was enabled to prove that the comets of 1682, of 1607, and
of 1531, were in reality so many successive apparitions of one and the
same body.