Abstract:
Let us consider the development of Greek stereometry during the 5th to 1th centuries BC. The focus is on the torus and curves on it including the Perseus curves and the Archytas curve. The torus is explicitly mentioned only by Perseus (2nd century BC) and Geminus of Rhodes (1st century BC) in the exposition of Heron of Alexandria (1st century AD) and Proclus Diadochus (5th century AD). However, one can assume that the torus had been known from the 4th century BC, but forgotten by Menelaus of Alexandria and Theon of Smyrna at the beginning of the 2nd century AD. Next, Eutocius of Ascalon (6th century AD) as well as Banu Musa brothers (9th century AD) also do not mention the torus. Of course, the torus serves as a natural part of the reconstruction of the Archytas solution to the problem of finding two mean proportional. On the other hand, curves on toroidal surfaces are useful for describing the motion of celestial bodies. Thus, the torus should have attracted geometers. So, from this point of view, we study the models of the world proposed by Heraclides of Pontus (4th century BC), Aristarchus of Samos (3rd century BC), and Hipparchus of Nicaea (2nd century BC). Later, Theon of Smyrna (2nd century AD) compared the motion of the planets to the spirals of a grapevine.
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