Elements of Astronomy (Alfraganus)

And the year contains 354 days, according to the absolute or round calculation, which is the more common one. But according to more subtle reasoning, this quantity of the Arabic year increases by eleven days over the course of thirty years, for which a portion of 1/3 and 1/30 of a day is yielded to each year. And thus the year precisely has 354 days, with 1/3 and 1/30 of one day: and that year, in which these fractions or parts constitute a whole day, claims seven full months and five hollow ones for itself.

However, this number corresponds to the days of the months, once the calculation is reduced to the conjunction of the Sun and Moon, according to the mean motion of both. But the number, starting from the first splendor of the new Moon, varies by excess or defect, such that subsequent months are not alternately full and hollow. For it does not always happen that the beginning of the month, by reason of both calculation and the first phase, falls on the same day; nor does this happen unless both are equalized between themselves over the passage of time.

The days of the Arabs, by which the months are counted, are seven days: the first of which is the day of the Sun, taking its beginning from the setting of the Sun on the Sabbath day; and its end from the setting of the same on the day of the Sun. The remaining days also behave in this manner. For the Arabs begin each day with its own night, that is