Autolycus, On a moving sphere Prop. 1©
translated by Henry Mendell (Cal. State U., L.A.)
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Prop. 1: If a sphere turns evenly about its own axis, all the points on the surface of the sphere which are not on the axis describe parallel circles having the same poles as the sphere, and, furthermore, upright to the axis.
(general diagram based on Vat. gr. 204 38v)
(diagram 1) Let there be a sphere of which let there be an axis, straight-line AB, and poles of it, points A, B, and let it turn evenly about the axis AB on it. I say that all the points on the surface of the sphere such as are on the axis describe parallel circles having the same poles as the sphere and furthermore upright to the axis.
(diagram 2) For let some point be taken on the surface of the sphere, G. And let there be drawn from G to straight-line AB a perpendicular, GD. (diagram 3) And let the plane through poles, A, B and GD be extended. In fact, it will make a section, a circle. (diagram 4) Let there be a semicircle of it, AGB. (diagram 5) If, in fact, with straight-line AB remaining-fixed, the semicircle in being rotated returns to the same spot from which it began to be moved, straight-line GD will also be rotated at every motion (μετακίνησις) of semicircle AGB, remaining-fixed at right-angles to straight-line AB, and will describe a circle in the sphere whose center will be point D, but the line from the center is GD, being at right-angles to axis AB. It is also obvious that points A, B will be poles of the described circle, since, in fact, a perpendicular has been from the center of the sphere and extended, AB, up to the surface of the sphere.
(diagram 6) We will, in fact, similarly show that all the points on the sphere such as are not on the axis, will also describe circles at right-angles to axis AB, having the same poles as the sphere. But circles about the same poles are parallel. Therefore, all the points on the surface of the sphere such as are not on the axis describe parallel circles having the same poles as the sphere, and, furthermore, upright to the axis.
Propositions paraphrased:
The plane through the axis of a sphere, being extended, makes a section, a circle.
If with a straight-line (the base of the semicircle) remaining-fixed in a sphere, a semicircle in the sphere, in being rotated (about the fixed straight-line), returns to the same spot from which it began to move, a straight-line perpendicular to the fixed straight-line (in the semicircle) in moving with the semicircle but remaining-fixed at right angles to the fixed straight-line will describe a circle in the sphere, whose line from the center (radius) is this line, whose center is the point that remains-fixed, and that is at right angles to the fixed straight-line.
Circles about the same poles are parallel. (this proposition clearly needs to be expanded)