The second type:
Circular dependence in legal rulings, as discussed in the Surayjiyya problem and elsewhere. We have devoted a separate work to this, explaining that it is invalid both rationally and under Islamic law. We have also addressed whether anything of this kind of circular dependence exists in the revealed Law or not.

(The third type:
Computational circular dependence: one says that this cannot be known until that is known. This is the kind that one seeks to resolve through arithmetic and algebra. We have explained that every question concerning the revealed Law brought by the Messenger, may Allah bless him and grant him peace, can be answered without algebraic calculation. Even if algebraic calculation is sound, we have explained that Islamic law and knowledge of it do not depend at all upon anything learned from non-Muslims, even if it is a valid method. Indeed, algebraic methods involve lengthy procedures for which Allah provides alternatives that make them unnecessary, as we have explained regarding logic.

The same applies to everything with which the Prophet, may Allah bless him and grant him peace, was sent, such as knowing the direction of the qiblah, the prayer times, the onset of dawn, and the appearance of the new crescent. All this can be known through the methods followed by the Companions and those who followed them in righteousness, without their needing anything else. Many people have nevertheless devised other methods, and many of them think that knowledge of the revealed Law is impossible without them. This stems from their ignorance.

Similarly, some people think that the direction of the qiblah cannot be known without knowing the longitudes and latitudes of places. Even if this is a sound mathematical science accessible to reason, Muslims’ knowledge of their qiblah does not depend upon it. Rather, it is authentically reported from the bearer of the revealed Law