The Sphere of the World (De Sphaera)

Definition of the sphere and certain geometric principles to be supposed.

In my opinion, regarding the interior astronomical guide?

I hope that which is above?

Definition of a sphere according to astronomy?

The sphere is described by Euclid as follows: A sphere is the passage of the circumference of a semicircle being carried around a fixed diameter until it returns to its place. That is, a sphere is such a round and solid object as is described by an arc of a semicircle being moved around. The sphere is also described by Theodosius as follows: A sphere is a certain solid contained by one surface in whose middle is a point, from which all lines drawn to the circumference are equal. And that point is called the center of the sphere. A straight line passing through the center of the sphere, applying its extremities to the circumference on both sides, is called the axis of the sphere. Two points indeed terminating the axis are called the poles of the world.

ON certain geometric principles to be supposed which have been added.

First sphere within the world; marginal note regarding... not a sphere... because of... in the hemisphere... the spheres are to...?

In the first place, those things which seem to be supposed and understood by those who do not know the discipline of geometry, by which they may more easily grasp what is to be said, are those which we subscribe below and have added for the sake of novices.

A diagram depicts a semicircle. The horizontal base is labeled "Diameter" and the curved arc is labeled "circumference of the semicircle." A vertical radius line is drawn from the center of the diameter to the arc.

A point in a thing having quantity is something indivisible, or of which there is no part that may be perceived by sight.
A line is length without breadth and depth, whose extremes are two points if it is understood as finite.
A straight line is the shortest extension from one extreme to the other.
A line that is not straight is one which curves between its extremes. The word "non" is abbreviated with an 'n' and a macron above the line.

A circular diagram with multiple radii drawn from the center, creating several segments. There is some partially illegible text written within and around the circle, including the word "degree."

A surface is length with breadth, lacking depth, which is terminated by a line or lines.
A plane surface is that which is extended most directly according to both length and breadth.
A non-plane surface is that which has some concavity or convexity.

A circular diagram represents a sphere with a vertical axis and horizontal diameter. The endpoints of the vertical axis are labeled "Pole." Text inside refers to the internal geometric properties.

An angle is the contact of two lines by a non-direct application.
A rectilinear angle is one which is made from two straight lines. A non-rectilinear angle is when one of the lines is not straight.

Three rectangular boxes contain geometric definitions and illustrations: Box 1 (Left) contains a straight horizontal line labeled "Straight line" and a wavy line labeled "Curved line." Box 2 (Center) contains a shaded trapezoidal shape labeled "Straight surface" and a shaded arch shape labeled "curved." Box 3 (Right) contains an acute angle with the word "angles" written below the vertex, and beneath it, two separate curved segments representing curved angles.