PHYSICS

small [circles] upon which the poles of the sphere revolve a great circle, and it is the circle T E K L, intersecting both circles C D and E Z at points E and C. And it is clear that point E is in the opposite of point C, [in] such an opposition, thus the line drawn between them will be the diameter of the sphere. And let us imagine the circle to be made by point A when the sphere moves to the motion of this diameter upon two circles E Z and C D, if the sphere has no other motion, to be the circle A F B; and then we will have a perfect circulation, as we declared in the preceding question. And thus let us imagine the circle to be made by point A when the sphere A B revolves upon poles K T, if the sphere were not moved by another motion, to be the circle A L B. And this circle, indeed, namely circle A L B, is by necessity opposite to both circles E Z and C D, and let it be placed upon the two points of intersection of these two great circles and upon poles T K, the horizon T A K.

A geometric diagram shows a circular structure representing the celestial sphere. Arcs labeled K, L, Q, A, S, E, C, D, T, and B intersect to demonstrate the motion of a point on a sphere moving on multiple axes simultaneously.

And when the ascent of point A is placed upon the horizon in the very place of the intersection of both circles, then when it has been compelled to its motion upon the circle of pole E to part Z, and the sphere has been declined to its compulsion, point A follows indeed on account of this toward the part of the declination of the pole. And when the sphere A B is moved further by another motion upon poles K T, point A will move from its place, declining indeed from the circle A F B, for it preserves the same distance always from pole E moved toward part Z, and it will come, for example, to point B. And when the sphere revolves upon poles T and K, and point B ascends from the horizon T A K, then it will not ascend from its first place but upon point Q of it, for example. And thus as long as pole E moves toward part Z, and when it has arrived at Z, for example, point A will arrive at L. Finally, in each revolution it will ascend from a point different from the point of its ascent in the succeeding revolution, and it will ascend from such a point and will finish its revolution at another point, and there will always be circles not of perfect circulation, for each one of them, namely of the circles, as we have said, is not in the same surface, and the figure of all will arrive at being the gyrating [figure] called laulabina spinning-top. And thus will be the disposition of point A when the pole moves from point Z to the point opposite to point E in the circle, and point A moves to its motion from L to B, for similarly a figure will be made similar to the first, and in the remaining two quarters there will be two figures similar to these two, and point A will return positioned to its first place. And thus the union of all two motions will be four figures, as we have declared, and this is what we wished to imagine.

And we say also that the circle marked by the stars from their motion upon their poles, whether fixed stars or planets, declines from the equinoctial according to the distance of the pole of that orb of the planet or of the fixed stars from the pole of the supreme [orb]—the pole, indeed, of the equinoctial—and the more the poles of the equinoctial are removed from the poles of the orb of the stars, the greater will be the declination of the circle in which the star will be, and [it is] as much as is the distance of the pole from the pole of the supreme [orb], and this is manifest by itself.

And thus we also say that when each of these orbs is moved by itself upon its poles toward the part of the motion of the universe, namely the daily motion, and a star fixed in that orb is moved to that motion by some space of its oblique circle, although we do not grasp this motion, then that which ascends from the degrees of that space which the star has traversed in its oblique circle will not always be equal to that which ascends with them from the degrees of the equinoctial circle, but sometimes it will be different according to the declination of that part which it passes of the circle of its declination from the equinoctial toward the north or toward the south, or upon the intersections themselves, and this has already been declared in the Almagest the great astronomical work of Ptolemy. And it has been shown also that...