Now some of the stars are seen neither to rise nor to set because they are borne on circles that are high up and are called "always visible" circles. These stars are those that come next to the visible pole and reach as far as the arctic circle [this is not a circle on the earth but rather one on the celestial shpere]. And, of these stars, those nearer the pole move on smaller circles, and those on the arctic circle on the greatest circle, the latter stars appearing actually to graze the horizon.
But all the stars which follow on these towards the south are all seen to rise and to set because their circles are not wholly above the earth, but part of them is above, and the remainder below, the earth. And of the segments of the several circles that are above the earth, that appears larger which is nearer to the greatest of the always-visible circles, while of the segments under the earth that which is nearer to the said circle is less, because the time taken by the motion under the earth of the stars which are on the said circle is the least, and the time taken by their motion above the earth is the greatest, while, for the stars on the circles which are continually further from the said circle, the time taken by their motion above the earth is continually less, and the time taken by their motion under the earth greater; the motion above the earth takes the least time, and the motion under the earth the greatest time, in the case of the stars which are nearest the south. The stars on the circle which is the middle one of all the circles appear to take equal times to complete their motion above the earth and their motion under the earth respectively, and hence we call this circle the "equinoctial" [this is the equator on the celestial sphere]; and those stars which are on circles equidistant from the equinoctial circle take equal times to describe the alternate segments; thus the segments above the earth in the northerly direction are equal to those under the earth in the southerly direction, and the segments above the earth in the southerly direction are equal to those under the earth in the northerly direction; but the sum of the times taken by the motion above the earth and by the motion under the earth continuous with it added together appears to be the same for each circle.
Further, the circle of the Milky Way and the zodiac circle, which are both obliquely inclined to the parallel circles and cut one another, appear in their revolution always to show semicircles above the earth.
On all the aforesaid grounds let us make it our hypothesis that the universe is spherical in shape; for if it had been in the form either of a cylinder or of a cone, the stars taken on the oblique circles bisecting the equinoctial circle would, in their revolution, have seemed to describe, not always equal semicircles, but sometimes a segment greater than a semicircle, and sometimes a segment less than a semicircle. For, if a cone or a cylinder be cut by a plane not parallel to the base, the section arising is a section of an acute-angled cone, which is like a shield (an ellipse). Now it is clear that, if such a figure be cut through its centre length-wise and breadth-wise respectively, the segments respectively arising are dissimilar; it is also clear that, even if it be cut in oblique sections through the centre, the segments formed are dissimilar in that case also; but this does not appear to happen in the case of the universe. For all these reasons, the universe must be spherical in shape, and revolve uniformly about its axis, one of the poles of which is above the earth and visible, while the other is under the earth and invisible.
Let the name "horizon" be given to the plane passing through our eye that is produced to the (extremities of the) universe, and separates off the segment which we see above the earth. The horizon is a circle; for, if a sphere be cut by a plane, the section is a circle.
Let the name "meridian circle" be given to the circle through the poles which is at right angles to the horizon, and the name "tropics" to the circles which are touched by the circle through the middle of the signs (the zodiac) and which have the same poles as the sphere.
The zodiac circle and the equinoctial circle are great circles; for they bisect one another. For the beginning of the Ram (Aries) and the beginning of the "Claws" (Libra) are diametrically opposite to one another and, both being on the equinoctial circle, the rising of the one and the setting of the other take place in conjunction, since they have between them six of the twelve signs of the zodiac, and two semicircles of the equinoctial circle, respectively, and since both beginnings, being on the equinoctial circle, take the same time to describe, the one its course above the earth, the other its course under the earth. But if a sphere rotates about its own axis all the points on the surface of the sphere describe, in the same time, similar arcs of the parallel circles on which they are carried; therefore the points in question traverse similar arcs of the equinoctial circle, on one side the arc above the earth, on the other the arc under the earth; therefore the arcs are equal; therefore both are semicircles, for the distance from rising to rising, or from setting to setting, is the whole circle; therefore the zodiac circle and the equinoctial circle bisect one another. But, if in a sphere two circles bisect one another, both of the intersecting circles are great circles; therefore the zodiac circle and the equinoctial circle are great circles.
The horizon, too, is one of the great circles. For it always bisects both the zodiac circle and the equinoctial circle; for it has always six of the twelve signs above the earth, and always a semicircle of the equinoctial circle above the earth; for the stars on the latter circle which rise and set respectively at the same time pass, in the same time, the one its course from rising to setting, the other its course from setting to rising. It is therefore manifest, from what was before proved, that there is always a semicircle of the equinoctial circle above the horizon. But if, in a sphere, a circle remaining fixed bisects any of the great circles that is moving continually, the circle which cuts it is also a great circle; therefore, the horizon is one of the great circles.
The time of a revolution of the universe is the time in which each of the fixed stars passes from one rising to its next rising, or from any place whatever to the same place again.
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