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Mathematical Treasure: Al-Amili's Essentials of Arithmetic

Frank J. Swetz (The Pennsylvania State University)

Bahāʾ al‐Dīn Muḥammad ibn Ḥusayn al‐ʿĀmilī, also known as Sheikh Bahā’ī ‎‎(1547-1621), was a Shīʿa Islamic scholar of the Safavid Empire in what is now Iran, and was fluent in both Persian and Arabic. Among his written works is Al-Khulāṣa fi’l isāb (Essentials of Arithmetic, sometimes instead translated as Quintessence of Calculation), a ten-chapter text in Arabic, which became a standard arithmetic in Central Asia and the Middle East for centuries. This copy of the original work was written in Mughal India in the Nastaʿlīq calligraphic style, which is more often used in Persian than in Arabic manuscripts. Note the added explanatory notes in the margins. These are the opening pages, with the author's full name appearing in the last two lines of the righthand page.

On the lefthand page below, just to the right of the red script, Al‐ʿĀmilī provided a list of the figures for the digits 1 through 9, read right to left. The symbol for zero, a dot, while not listed here, is used throughout the manuscript, including in the lefthand page in the next image.

The examples concerning the operation of addition are recognizable to a modern reader. On the lefthand page below, the calculation near the top shows an example in which six 6-digit numbers are added together. The units digit of the sum is 0, signified in Arabic numeration by a dot.

Al‐ʿĀmilī provided a “Table of Multiplication Facts” in triangular form acknowledging the commutative property of multiplication. The digits 2 through 9 run down the righthand side of the table.

Graphically, the author introduced his readers to several algorithms for the multiplication of two multi-digit factors.

These images are obtained through the courtesy of the World Digital Library, where the entire manuscript can be viewed. The five "pages" shown above are numbered 3, 4, 5, 6, and 9, with each divided into a righthand page (e.g. 3R) and a lefthand page (e.g. 3L) in that order since Arabic is written and read from right to left. If you view the manuscript at the World Digital Library website, note that page 2L is the title page. The manuscript itself is held by the Qatar National Library.

Source: The Essentials of Arithmetic (al-Khulasa fil hisab) via World Digital Library:

Acknowledgment: The author and editor thank Randy Schwartz for his contributions to this article.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Al-Amili's Essentials of Arithmetic," Convergence (July 2017)