There has, however, been handed down a set of numbers attributed to him giving the distances of the various heavenly bodies from Earth, which has been shown to be based not on observed astronomical data but on a “Pythagorean” theory associating the spatial intervals between the planets with musical intervals. Those of his results that could be simply expressed such as the formulas for the surface area and volume of a sphere became mathematical commonplaces, and one of the bounds he established for, 22/7, was adopted as the usual approximation to it in antiquity and the Middle Ages. Nevertheless, his mathematical work was not continued or developed, as far as is known, in any important way in ancient times, despite his hope expressed in Method that its publication would enable others to make new discoveries.
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