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A New Role for the Hippopede of Eudoxus

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Abstract

The geometry of the alternative reconstruction of Eudoxan planetary theory is studied. It is shown that in this framework the hippopede acquires an analytical role, consolidating the theorys geometrical underpinnings. This removes the main point of incompatibility between the alternative reconstruction and Simpliciuss account of Eudoxan planetary astronomy. The analysis also suggests a compass and straight-edge procedure for drawing a point by point outline of the retrograde loop created by any given arrangement of the three inner spheres.

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... Increasingly accurate, but also ever more cumbersome, models were developed by Callip tion in order to prevent the rotation of one planet's spheres from influencing another's; only the moon lacks such retroactive spheres, as it is situated nearest the earth, at the center of the model (Gregory, 2016, pp. 104-107; see also Mendell, 1998;Yavetz, 1998Yavetz, , 2001. ...
... This implies that the intersection curve is a figure-eight loop with a self-intersection at s 0+ (see figure 3 (middle)). This curve is also known as the Hippopede of the Greek astronomer and mathematician Eudoxus of Cnidus (408 BC – 355 BC) [14, 18]. In the Bose-Hubbard mean-field dynamics it appears as a separatrix curve, which separates the flow inside (for smaller values of r) from the flow outside (for larger values of r), (c) In the most singular situation cases (a) and (b) coincide, i.e. the cylinder passes through the center and touches the sphere. ...
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Sometime in the first half of the fourth century B.C., EUDOXUS of Cnidos attempted to recreate the apparent motions of the sun, the moon, and the five thenknown planets by certain arrangements of nested spheres rotating concentrically with the celestial sphere, the earthbound observer being located at the common centre. Modern commentators have studied the astronomical career of these mathematical models, with particular interest in the precise extent to which the models might be able to recreate the phenomena, in the refinements added to them