One reason why some Greeks had problems accepting irrationals is that the existence of intervals in irrational ratios to each other defeated the assurance that all cosmic cycles have a common unit and thus that all configurations of the cosmos recur eternally.
Interesting. Can you provide a reference in antiquity for such thoughts?
Dear Jon, I find this difficult to believe. The Greeks lasted for hundreds of years. What period are we talking about? There were various models of the cosmos in Greek science. Even after Ptolemy, they knew the difficulty of predicting the positions of the planets. They did believe that circular motion was eternal. Thus some could not accept Ptolemy's equant. Please take the time to find a reference. I am interested in this problem. Peace, Don