The Sphere of the World

...is the circle dividing the lower hemisphere from the upper. Whence it is called the horizon the boundary between the earth and the sky, that is, the "limiter of sight." It is also called the circle of the hemisphere. There are two types of horizons: the right and the oblique (or sloping). Those who have their zenith the point directly overhead on the equinoctial the celestial equator possess a right horizon and a right sphere. This is because their horizon is a circle passing through the poles of the world the north and south celestial poles, dividing the equinoctial at right spherical angles. For this reason, it is called a "right horizon" and a "right sphere."

Conversely, those for whom the pole of the world is elevated above the horizon have an oblique or sloping horizon, because their horizon intersects the equinoctial at unequal and oblique angles. Whence it is called an oblique horizon and an oblique or sloping sphere.

The zenith above our heads is always the pole of our horizon. From this, it is clear that however much the pole of the world is elevated above the horizon, the distance of the zenith from the equinoctial is exactly the same. This is demonstrated as follows: Since on any natural day each colure a great circle passing through the celestial poles is joined twice to the meridian (or is essentially the same as the meridian), whatever is proven about one applies to the other. Let us, therefore, take a quarter part of the colure that distinguishes the solstices, which runs from the equinoctial to the pole of the world. Then, take another quarter part of the same colure that runs from the zenith to the horizon (since the zenith is the pole of the horizon). Because these two segments are both quarters of the same circle, they are equal to one another. According to the rules of geometry, if equal parts—or a shared common part—are subtracted from equals, the remainders will be equal. Therefore, if we subtract the common arc The shared segment of the circle between the zenith and the celestial pole that lies between the zenith and the pole of the world, the remaining parts will be equal: namely, the elevation of the pole of the world above the horizon and the distance of the zenith from the equinoctial.

¶ Concerning the four minor circles.

Having spoken of the six major circles The Horizon, Meridian, Equinoctial, Zodiac, and the two Colures, we must now speak of the four minor ones. It should be noted that when the sun is in the first point of Cancer (the point of the summer solstice), it describes a certain circle through the daily rotation of the firmament. This circle is the furthest one described by the sun toward the arctic northern pole. For this reason, it is called the "circle of the summer solstice," as explained above, or the summer tropic. It takes its name from the Greek word tropos original: "tropos", meaning a turn or change in direction, which signifies a "turning," because at that point the sun begins to turn back toward the lower hemisphere.