Likewise, the sun’s alignment with this or that sign of the zodiac cannot be perceived by sight. It can only be determined through obscure, specialized, difficult calculations that are liable to error. Through sensory perception, it can be known only approximately.

When winter ends and the season the Arabs call “summer,” which people call “spring,” begins, this is the time when the sun reaches the equinoctial point at the beginning of Aries. The same applies in autumn. What can be perceived through the senses, then, is winter and summer, and the two equinoxes between them, approximately. As for the sun’s passage from one sign to another, this can be known only through calculations that require effort and distract from other pursuits, despite their limited benefit. It is thus clear that the crescent alone provides an observable, universally recognizable boundary for determining appointed times.

The customs of nations in reckoning their months and years fall into all the logically possible categories: the month and year may both be defined numerically, or both by natural cycles; alternatively, the month may follow a natural cycle and the year be defined numerically, or vice versa.

Those who define both numerically include those who make the month thirty days and the year twelve months. Those who define both by natural cycles include those who make the month lunar and the year solar, adding at the end of the months the days accounting for the difference between