There is no doubt that the authentic Sunnah and the agreement of the Companions establish that it is impermissible to rely on astronomical calculations. As authentically reported from the Prophet in the two Sahīh collections, he said:

“We are an unlettered nation: we neither write nor calculate. Begin fasting upon sighting the crescent, and end fasting upon sighting it.”

Whoever relies on calculations to establish the crescent is not only astray in matters of Islamic law and an innovator in religion, but also mistaken in his reasoning and understanding of calculation. Astronomers know that visibility cannot be precisely determined by calculation. At most, their calculations, when accurate, can establish how many degrees separate the crescent from the sun at sunset, for example. Visibility, however, is not governed by fixed numbers of degrees. It varies according to the sharpness or weakness of eyesight, the elevation of the place from which the crescent is sought, and the clarity or murkiness of the atmosphere. One person may see it at a separation of eight degrees, while another may fail to see it at twelve degrees.

For this reason, those versed in calculation have disagreed widely and inconsistently over the arc of visibility. Their leading authorities, such as Ptolemy, did not say a single word on the subject, because it cannot be established by mathematical proof. Only some of their later scholars, such as Kūshyār al-Daylamī and others like him, discussed it. When they saw that Islamic law made its rulings contingent on the crescent, they regarded calculation as a method by which to determine precisely