“The fixed stars rise at the same point, and set at the same point; the same stars always rise together, and set together, and in their course from the east to the west they always preserve the same distance from one another. Now, as these appearances are only consistent with a circular movement, when the eye of the observer is equally distant from the circumference of the circle in every direction (as has been demonstrated in the treatise on Optics), it follows that the stars move in a circle and are attached to a single body, and that the vision is equally distant from the circumference. [Illustration: FIG. 3.—Illustration of Euclid’s statements. _P_ the star between the Bears. _D D´_ the region of the always visible. _C B A_ the regions of the stars which rise and set. ] “A star is visible between the Bears, not changing its place, but always revolving upon itself. Since this star appears to be equally distant from every part of the circumference of each circle described by the other stars, it must be assumed that all the circles are parallel, so that all the fixed stars move along parallel circles, having this star as their common pole. “Some of these neither rise nor set, on account of their moving in elevated circles, which are called the ‘always visible.’ They are the stars which extend from the visible pole to the Arctic circle. Those which are nearest the pole describe the smallest circle, and those upon the Arctic circle the largest. The latter appears to graze the horizon. “The stars to the south of this circle all rise and set, on account of their circles being partly above and partly below the earth. The segments above the earth are large and the segments below the earth are small in proportion as they approach the Arctic circle, because the motion of the stars nearest the circle above the earth is made in the longest time, and of those below the earth in the shortest. In proportion as the stars recede from this circle, their motion above the earth is made in less time, and that below the earth in greater. Those that are nearest the south are the least time above the earth, and the longest below it. The stars which are upon the middle circle make their times above and below the earth equal; whence this circle is called the Equinoctial. Those which are upon circles equally distant from the equinoctial make the alternate segments in equal times. For example, those above the earth to the north correspond with those below the earth to the south; and those above the earth to the south correspond with those below the earth to the north. The joint times of all the circles above and below the earth are equal. The circle of the milky way and the zodiacal circle being oblique to the parallel circles, and cutting each other, always have a semicircle above the earth.