The Sphere of the World

...neither beginning nor end. From this, in likeness to it, the sensible world original: "mundus sensibilis"; the physical universe perceived by the senses has a round shape, in which one cannot assign a beginning or an end.

Convenience: because of all isoperimetric bodies: different shapes that have the same surface area, the sphere is the largest; also, of all forms, the round is the most capacious. Since, therefore, it is the largest and round, it is the most capacious. Thus, since the world contains all things, such a shape was useful and convenient for it.

Necessity: because if the world were of another shape besides round—namely three-sided, four-sided, or many-sided—two impossibilities would follow: namely, that some place would be empty, and some body would be without a place, both of which are false, as is evident in the case of raised and rotated angles.

Also, as Alfraganus Ahmad ibn Muhammad ibn Kathir al-Farghani, a 9th-century Persian astronomer whose works were a primary source for medieval European astronomy says, if the heaven were flat, some part of the heaven would be closer to us than another—specifically, the part directly over our heads. Therefore, a star located there would be closer to us than one at its rising or setting. But those things which are closer to us appear larger. Thus, the sun or another star in the middle of the sky should appear larger than when rising or setting; yet we see the opposite happening. For the sun or another star appears larger when it is in the east or west than in the middle of the sky.

But since the truth of the matter is not so, the cause of this appearance is that in winter or rainy weather, certain vapors rise between our sight and the sun or another star; and since those vapors are a diaphanous body: a transparent or translucent substance, they disperse our visual rays so that they do not perceive the thing in its natural and true size. This is evident from a penny original: "denario"; a common silver coin tossed into the bottom of clear water, which, due to a similar dispersion of rays, appears larger than its true size.

That the earth is round.

That the earth is also round is shown thus: the signs and stars do not rise and set at the same time for all people everywhere. Instead, they rise and set earlier for those who are toward the east; and the cause for them rising and setting sooner for some and later for others is the bulge of the earth original: "tumor terre"; referring to the curvature of the Earth's surface.