understand them completely it is necessary to trace them to their
origin. The problem of calculating the motions of the heavenly
bodies and the changes in the aspect of the celestial sphere was
one of the first with which the students of astronomy were
occupied. Indeed, in ancient times, the only astronomical problems
which could be attacked were of this class, for the simple reason
that without the telescope and other instruments of research it
was impossible to form any idea of the physical constitution of
the heavenly bodies. To the ancients the stars and planets were
simply points or surfaces in motion. They might have guessed that
they were globes like that on which we live, but they were unable
to form any theory of the nature of these globes. Thus, in The
Almagest of Ptolemy, the most complete treatise on the ancient
astronomy which we possess, we find the motions of all the
heavenly bodies carefully investigated and tables given for the
convenient computation of their positions. Crude and imperfect
though these tables may be, they were the beginnings from which
those now in use have arisen.

No radical change was made in the general principles on which
these theories and tables were constructed until the true system
of the world was propounded by Copernicus. On this system the
apparent motion of each planet in the epicycle was represented by
a motion of the earth around the sun, and the problem of
correcting the position of the planet on account of the epicycle
was reduced to finding its geocentric from its heliocentric
position. This was the greatest step ever taken in theoretical
astronomy, yet it was but a single step. So far as the materials
were concerned and the mode of representing the planetary motions,
no other radical advance was made by Copernicus. Indeed, it is