# Mathematical Treasure: Ibn al-Banna's Operations of Calculation

Author(s):
Frank J. Swetz (The Pennsylvania State University)

During the reign of the Berber-Muslim dynasties of North Africa and Andalusia (now southern Spain), study of the sciences was eclipsed by the study of mathematics, and noteworthy mathematical works survive from this period. The Lifting of the Veil in the Operations of Calculations, written by Abū'l Abbās Ahmad ibn al-Bannāʾ (1256-1321) of what is now Morocco, demonstrates a high level of mathematics at this time. The Lifting of the Veil is a commentary by the author on his own earlier work, The Abridgement of the Operations of Calculation, a text which may have proved too difficult for his readers. In The Lifting of the Veil, ibn al-Bannāʾ described in detail computations involving arithmetic series, continued fractions, combinatorics, and the use of binomial coefficients.

Figure 1. Title page of The Lifting of the Veil in the Operations of Calculations

The Desire of the Students for an Explanation of the Calculator’s Craving was compiled by Muhammad ibn Gāzī (ca. 1437-1513), who lived in what is now Morocco. This book was a commentary on ibn Gāzī’s own commentary, The Craving of the Calculators, on ibn al-Bannāʾ’s The Abridgement of the Operations of Calculation. In The Desire of the Students, passages in red ink are from ibn Gāzī’s first commentary, The Craving of the Calculators, and those in black ink were new to The Desire of the Students. The images shown in Figures 2 and 3 are believed to be from an 18th century copy of the work.

Figure 2. First page of text of The Desire of the Students for an Explanation of the Calculator’s Craving

Figure 3. Examples of the gelosia method of multiplication using both right and left slanting diagonals within the grids in The Desire of the Students for an Explanation of the Calculator’s Craving

##### References

The Lifting of the Veil in the Operations of Calculations, World Digital Library: http://www.wdl.org/en/item/4255/

The Desire of the Students for an Explanation of the Calculator’s Craving, World Digital Library: http://www.wdl.org/en/item/4257/

##### Acknowledgment

The images above were obtained through the courtesy of the World Digital Library (WDL) and the assistance of the digital imaging staff at the United States Library of Congress. All items can be viewed in full at the WDL website.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Ibn al-Banna's Operations of Calculation," Convergence (January 2014)