The Sphere of the World with Commentaries

FOR NOVICE YOUTHS: For those desiring an approach to the study of astronomy, and for those undertaking the affairs of the republic, this short and direct path, removed from the common track, begins with the Spherical Tractate of Johannes de Sacrobosco, the most accurate Theories of the Planetary Motions by Georg Purbach, and also the most accurate and useful Disputations of Johannes Regiomontanus against the Cremonensian ravings concerning the theories of these same planets—a work compiled in a useful sequence.

We distinguish the Treatise on the Sphere into four chapters. First, we shall speak of what a sphere is, what its center is, what the axis of a sphere is, what the pole of the world is, how many spheres there are, and what the form of the world is. In the second, of the circles from which the material sphere is composed, and which that supercelestial [sphere] is understood to be composed, which is imagined through this one. In the third, of the rising and setting of the signs, of the diversity of days and nights that occurs for those dwelling in different places, and of the division of the climates. In the fourth, of the circles and motions of the planets, and of the causes of eclipses.

CHAPTER ONE.

A sphere, therefore, is described by Euclid thus: A sphere is the passage of the circumference of a semicircle, as often as it is carried around a fixed diameter until it returns to its own place. That is to say, a sphere is such a round and solid object that is described by the arc of a semicircle being carried around. A sphere is also described by Theodosius thus: A sphere is a certain solid contained by one surface, in the middle of which 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 indeed passing through the center of the sphere, applying its extremities to the circumference on each side, is called the axis of the sphere. The two points indeed terminating the axis are called the poles of the world.

[Diagram labels: Circumference of the semicircle; Fixed diameter]