Principles of Hindu Reckoning

Principles of Hindu Reckoning (Arabic: كتاب في أصول حساب الهند, romanized: Kitab fi usul hisab al-hind) is a mathematics book written by the 10th- and 11th-century Persian mathematician Kushyar ibn Labban. It is the second-oldest book extant in Arabic about Hindu arithmetic using Hindu-Arabic numerals ( ० ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹), preceded by Kitab al-Fusul fi al-Hisub al-Hindi (Arabic: كتاب الفصول في الحساب الهندي) by Abul al-Hassan Ahmad ibn Ibrahim al-Uglidis, written in 952.

Although Al-Khwarizmi also wrote a book about Hindu arithmetic in 825, his Arabic original was lost, and only a 12th-century translation is extant. In his opening sentence, Ibn Labban describes his book as one on the principles of Hindu arithmetic. Principles of Hindu Reckoning was one of the foreign sources for Hindu Reckoning in the 10th and 11th century in India. It was translated into English by Martin Levey and Marvin Petruck in 1963 from the only extant Arabic manuscript at that time: Istanbul, Aya Sophya Library, MS 4857 and a Hebrew translation and commentary by Shālôm ben Joseph 'Anābī.

Hindu arithmetic was conducted on a dust board similar to the Chinese counting board. A dust board is a flat surface with a layer of sand and lined with grids. Very much like the Chinese counting rod numerals, a blank on a sand board grid stood for zero, and zero sign was not necessary. Shifting of digits involves erasing and rewriting, unlike the counting board.

There is only one Arabic copy extant, now kept in the Hagia Sophia Library in Istanbul. There is also a Hebrew translation with commentary, kept in the Bodleian Library of Oxford University. In 1965 University of Wisconsin Press published an English edition of this book translated by Martin Levey and Marvin Petruck, based on both the Arabic and Hebrew editions. This English translation included 31 plates of facsimile of original Arabic text.

Principles of Hindu Reckoning consists of two parts dealing with arithmetics in two numerals system in India at his time.

The similarity between decimal Hindu algorithm with Chinese algorithm in Sunzi Suanjing are striking, except the operation halving, as there was no hybrid decimal/sexagesimal calculation in China.

Kushyar ibn Labban described in detail the addition of two numbers.

The Hindu addition is identical to rod numeral addition in Sunzi Suanjing