heavens. There are stars over your head, and half the contents of
the heavens are visible, while the other half are below your
horizon. As the earth turns round, the stars over your head will
change, and unless it should happen that you have taken up your
position at either of the poles, new stars will pass into your view,
and others will disappear, for at no time can you have more than half
of the whole sphere visible. The observer on the earth would,
therefore, say that some stars were rising, and that some stars were
setting. We have, therefore, two totally distinct methods, each of
which would completely explain all the observed facts of the diurnal
movement. One of these suppositions requires that the celestial
sphere, bearing with it the stars and other celestial bodies, turns
uniformly around an invisible axis, while the earth remains
stationary at the centre. The other supposition would be, that it is
the stupendous celestial sphere which remains stationary, while the
earth at the centre rotates about the same axis as the celestial
sphere did before, but in an opposite direction, and with a uniform
velocity which would enable it to complete one turn in twenty-four
hours. Ptolemy was mathematician enough to know that either of these
suppositions would suffice for the explanation of the observed
facts. Indeed, the phenomena of the movements of the stars, so far
as he could observe them, could not be called upon to pronounce which
of these views was true, and which was false.

Ptolemy had, therefore, to resort for guidance to indirect lines of
reasoning. One of these suppositions must be true, and yet it
appeared that the adoption of either was accompanied by a great
difficulty. It is one of his chief merits to have demonstrated that
the celestial sphere was so stupendous that the earth itself was
absolutely insignificant in comparison therewith. If, then, this