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I'm having a hard time gathering my thoughts about this. I'm trying to find a connection or some sort of relation between the first 3 axioms (postulates) of Euclidean geometry (though around Aristotle's time it wasn't called Euclidean geometry yet, but the concepts were there) and Aristotle's theory of elements and motion.

I know that Aristotle believed that elements move on a straight line according to their 'weight' (heavy to light), and that objects have 2 natural movements, towards the center of the earth (downwards) or further from the center of the earth (upwards). So that I think kind of covers the first geometrical axiom, of a straight line between 2 points. What about the other two?

I'm aware of his adoption of Eudoxus geometrical planetary movements model and his uniform circular motion of the planets, but I can't seem to think of a way to connect them in any way to the geometrical axioms.

Any ideas, guys?

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    $\begingroup$ No specific link... You can see the pseudoi-Aristotelian treatise Mechanica for a "geometrical" treatment of some topics regarding motion. $\endgroup$ – Mauro ALLEGRANZA Jan 30 at 7:47
  • $\begingroup$ In Aristotle's physics natural motions in the sublunar sphere are in straight lines towards the center of the Earth, and in the superlunar sphere (heavens) are circles around the said center. Sublunar motions can be stopped and resumed, superlunar ones are eternal. You can connect that to the straightedge and compass constructions of segments, lines, and circles in postulates 1-3, I suppose. $\endgroup$ – Conifold Jan 30 at 19:00
  • $\begingroup$ I was thinking more of the link @Conifold has mentioned. Thanks a lot, guys! $\endgroup$ – j4dk Feb 2 at 8:24
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On the Heavens, bk.1 (chap.2) , is perhaps the place to read carefully. Simple motions are of two types, radial and circular, and combinations of them produce any other motion. ""Radial" motion is further described as to the center and from the center, while the movement around a center is "without an opposite". Circular motion pertains to the first element, which later somehow became the fifth one (quint-essence). It is inexhausible, eternal and does not need a mover, at least in this work. Following Eudoxus, Aristotle devised a mechanical model with one self rolling outer sphere whose movement is transmitted through a complicated system of homocentric smaller spheres down to the lunar sphere. The world being finite, there are not actually arbitrarily long lines. Physics is the science about Nature (physis), and it is definitely not Geometry.

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