The Sphere of the World (De Sphaera)

A circular diagram in the top-left margin showing a circle with a wedge-shaped section (a sector) cut into it. The two sections of the circle are labeled "scin" and "da" (fragmentary labels for "scinda").

If two straight lines extended from the center to the circumference create an angle over the center, it has pleased some to call the enclosed part of the circle a scinda sector/cut portion of the circle. It should be noted that it is frequently in use that the circumference is taken for the circle, which craftsmen commonly observe, and so we shall observe it when it is pleasing.

Note

If a surface is enclosed from three lines, a three-sided figure is made which is also called a trigon and a triangle. If from four, it is a quadrilateral or a tetragon; and from five, a pentagon; and the others are similarly denominated from the number of sides or angles. Each flat figure indeed has as many sides as it has angles, provided it has sides; since a circle has no sides. But if all sides are consistent and equal, all its angles will be equal to each other, which conversely is not always the case, especially in a long and rectangular quadrilateral. This contains all angles equal, because they are right angles, but only the opposite sides are equal. That flat figure is to be called rectilinear which is completed on all sides by straight lines.

In the left margin, a set of geometric figures: a square labeled "square," a triangle labeled "triangle," a pentagon labeled "pentagon," and a rectangle labeled "long quadrilateral."

A sphere is the first figure among bodies, just as a circle is among surfaces. A sphere, according to Theodosius, is a corporeal figure contained by one surface, within which is a point from which all straight lines led out that meet that surface are equal among themselves; and that point is the center of the sphere. From which it is clear that a sphere is a round body, round with a perfection of roundness according to geometers.

A sphere in general is called... in one species... poles are two... of the world... circumference... poles are fixed points... which... ?

A straight line which stretches through the center of a sphere and touches its surface on both sides is not properly to be called the diameter of the sphere, but as many call it, a menguar axis/measure, because although it penetrates through the middle of the sphere, it does not divide it through the middle, but is more properly called a menguar or axis, especially if the sphere revolves around it while it remains fixed. And two opposite points on the surface of the sphere terminating the axis are called poles.

Two circular diagrams at the bottom left. The left one represents a sphere with a vertical axis labeled "Axis" and top/bottom poles labeled "Pole." The right one shows a circle with several diameters intersecting at the center like spokes of a wheel.

Every circle which cuts a sphere into two halves deserves to be called the diameter of the sphere and is called a great circle of the sphere; and in the same sphere, all great circles are equal because they pass through the center of the sphere; wherefore all such are concentric; furthermore, every two great circles in a sphere divide each other into equal parts.

Every circle cutting a sphere, whose periphery is revolved along its surface, is called a sector of the sphere. And each such has its own axis which runs through the center of the sphere, upon which the center of that circle always insists, and the extreme points of that same axis are the poles of that circle. But if it is a great circle, the poles will distance themselves from its circumference by an equal measure. If, however, it is a smaller one, one of them will distance itself more and the other less from the same, provided that each one of them always remains equidistant from it.