Complete Works of Claudius Ptolemy, Vol. II

so that they are, as it were, joined together without an interval, not disturbed by the rotations of the other spheres around different axes. The essence of the method for accomplishing this appears sufficiently in Figure 1, which illustrates the motion of the fixed stars: the sphere of the fixed stars $γε$ is connected with the first mover $αγ$ at the poles $γ, δ$, and the second mover $εη$ is attached to $γε$ at the poles $ε, ζ$. Now, if both axes $γδ$ and $εζ$ are perpendicular to the plane of the zodiac, those two movers can be carried around the axis of the universe $αη\varthetaβ$ as if they were joined together without an interval, while the sphere of the fixed stars, unimpeded by them, rotates around the axis $γζ$, the motion by which precession is explained. Similarly, the rotations of the other movers around the axis of the universe become independent and free from them, which produces the proper motions of the seven planets.

Besides this method, Ptolemy also explains another, where parts of spheres are substituted for the spheres themselves. As far as the mathematical ratio is concerned, it differs little from the former.

Having enumerated all the spheres or parts of spheres that the entire system contains according to those two methods, Ptolemy explains the use of certain tables which were undoubtedly added at the end of the book. With their help, the positions of the planets could be computed for any given moment in time by using numbers taken from individual tables, which would indicate the state of each sphere at that time.