I mentioned above that al-Karajī and Ibn al-Hāʾim both wrote proofs for finger-reckoners, the people al-Karajī described as "understanding by means of the tongue and hand." Ibn al-Bannāʾ's proofs based on completing the square in his algebra book would have been accessible to that crowd, too. Ironically, his proofs based in a rule originating in finger-reckoning were not meant for finger-reckoners. In Ibn al-Bannāʾ's hands the "squaring" rule migrated from a practical way of mentally multiplying numbers to the foundation for novel proofs in algebra. Like the high-level arithmetic books of other authors like Ibn Munʿim (12th-13th c.) and al-Fārisī, Ibn al-Bannāʾ's *Lifting* *the Veil* was intended for a more theoretically-minded readership.

In medieval Islam there was recurrent interplay between the different practical and theoretical traditions in mathematics. To pick three examples,

- al-Khayyām adapted practical algebra for use in Greek-style geometrical problem-solving,
- al-Fārisī gave the rules of finger-reckoning, including the "squaring" rule, a foundation in proofs modeled on those in Euclid's
*Elements*Books VII-IX, and - an anonymous ninth-century text, probably written by al-Māhānī, applied algebra to find numerically the roots of the binomials and apotomes described in Book X of Euclid's
*Elements*.

Ibn al-Bannāʾ's *Lifting* *the Veil* is another example of this merging of the mathematics originating in different contexts.

Jeffrey Oaks received his Ph.D. in mathematics from the University of Rochester in 1991, and he has taught mathematics at the University of Indianapolis since 1992. He began the study of history of mathematics in 1999, and has since published many studies in history of algebra, particularly in medieval Arabic algebra.