In the Late Byzantine era (late 13th -15th c.), specially in Constantinople, the capital city of the Byzantine Empire, we could find few scholars, who wrote about astronomy following the Ptolemaic tradition. Conversation about this mathematical science was very popular between scholars, and the main subject was renovation of astronomical data. In the same period, in Mongolian Persia, near the Byzantine Trebizond, there were healthy observatories at Maraga and Tabriz, where scholars followed the Islamic tradition, and in Provence the Jewish common of Caraites created a new astronomical school, which was based in Hebrew rules.
Soon, the basic principles of Persian and Jewish traditions were transferred in Constantinople. In these conditions, three teams were created about astronomy: first, the scholars who followed the Ptolemaic astronomy and believed that the solution of the problems was the right study of ancient Greek corpus, second, the scholars who imported principles of Persian astronomical texts, and, third, the scholars who accorded with the principles of Caraites.
In this paper we will present, first, the transmission of ideas, second, the three schools with their representatives, and, third, the debate between them. Then, we will try to interpret the main arguments in this conflict and the ideology behind them, and at least, the effects of this fruitful conversation.
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