The Sphere of the World with Commentaries

is the circle dividing the inferior hemisphere from the superior. Whence it is called the horizon, that is, the terminator of the view. It is also called the circle of the hemisphere. The horizon is, however, twofold: right and oblique or declivous. Those whose zenith is on the equinoctial have a right horizon and a right sphere, because their horizon is a circle passing through the poles of the world, dividing the equinoctial at right spherical angles, whence it is called a right horizon and a right sphere. Those for whom the poles of the world are elevated above the horizon have an oblique or declivous horizon, because their horizon intersects the equinoctial at unequal and oblique angles; whence it is called an oblique horizon and an oblique or declivous sphere. The zenith of our head is always the pole of the horizon. Whence it is evident from these that the elevation of the poles of the world above the horizon is equal to the distance of the zenith from the equinoctial. Which is evident thus: Since in every natural day each of the colures joins the meridian twice, or is the same as the meridian, whatever is proved of one is proved of the other. Let the fourth part of the colure distinguishing the solstices, which is from the equinoctial to the pole of the world, be taken. Let the fourth part of the same colure, which is from the zenith to the horizon, be taken again, since the zenith is the pole of the horizon. These two quarters, since they are quarters of the same circle, are equal to each other. But if from equal things equal things be removed, or the same common thing, the remainders will be equal. Having removed, therefore, the common arc—namely, that which is between the zenith and the pole of the world—the remainders will be equal, namely: the elevation of the poles of the world above the horizon and the distance of the zenith from the equinoctial.

CONCERNING THE FOUR SMALLER CIRCLES.

Having spoken of the six greater circles, we must speak of the four smaller ones. It is to be noted, therefore, that the sun, existing in the first point of Cancer, or in the point of the summer solstice, by the rapid movement of the firmament describes a certain circle which is the last described by the sun from the side of the arctic pole. Whence it is called the circle of the summer solstice for the reason stated above, or the summer tropic, from tropos, which is conversion, because then the sun begins to turn itself toward the inferior hemisphere and to recede from us. The sun again, existing in the first point of Capricorn or of the winter solstice,