globular form
of the earth, the explanation of the moon's phases, and indeed all the
successive steps taken, involved this same mental process. But we must
content ourselves with referring to the theory of eccentrics and
epicycles, as a further marked illustration of it. As first suggested,
and as proved by Hipparchus to afford an explanation of the leading
irregularities in the celestial motions, this theory involved the
perception that the progressions, retrogressions, and variations of
velocity seen in the heavenly bodies, might be reconciled with their
assumed uniform movement in circles, by supposing that the earth was not
in the centre of their orbits; or by supposing that they revolved in
circles whose centres revolved round the earth; or by both. The
discovery that this would account for the appearances, was the discovery
that in certain geometrical diagrams the relations were such, that the
uniform motion of a point would, when looked at from a particular
position, present analogous irregularities; and the calculations of
Hipparchus involved the belief that the relations subsisting among these
geometrical curves were _equal_ to the relations subsisting among the
celestial orbits.

Leaving here these details of astronomical progress, and the philosophy
of it, let us observe how the relatively concrete science of geometrical
astronomy, having been thus far helped forward by the development of
geometry in general, reacted upon geometry, caused it also to advance,
and was again assisted by it. Hipparchus, before making his solar and
lunar tables, had to discover rules for calculating the relations
between the sides and angles of triangles--_trigonometry_ a subdivision
of pure mathematics. Further, the reduction of the doctrine of the
sphere to the quantitative form needed for astronomical purposes,
required the formation of a _spherical trigonometry_, which was also
achieved by Hipparchus. Thus both plane and spherical trigonometry,
which are parts of the highly abstract and simple science of extension,
remained undeveloped until the less abstract and more complex science of
the celestial motions had need of them. The fact admitted by M. Comte,
that since Descartes the progress of the abstract division of
mathematics has been determined by that of the concrete division, is
paralleled by the still more significant fact that even thus early the
progress of mathematics was determined by that of astronomy.

And here,