AL-BAGHDāDī, ABū MANṢūR ʿABD AL-QāHIR IBN ṬāHIR IBN MUḤAMMAD IBN ʿABDALLAH, AL-TAMīMī, AL-SHAFIʿī

(b. Baghdad; d, 1037)

arithmetic.

The last two names indicate the tribe from which Abū Mansūr was descended and the religious school to which he belonged. Born and raised in Bagdad, he left with his father for Nīshāpūr (or Nīshābūr), taking with him great wealth that he spent on scholars and scholarship. Riots broke out in Nīshāpūr, and he moved to the quieter town of Asfirāyīn. His departure was considered a great loss to Nīshāpūr. In his new home, he continued to pursue learning and to propagate it. He is reported to have lectured for years in the mosque, on several subjects, never accepting payment. Although he was one of the great theologians of his age, and many works are attributed to him, none has been studied scientifically. We are concerned here with two works on arithmetic.

The first is a small book on mensuration, Kitāb fi’l-misāha, which gives the units of length, area, and volume and ordinary mensurational rules.

The second, al-Takmila fi’l-hisāb, is longer and far more important. In the introduction Abū Mansūr notes that earlier works are either too brief to be of great use or are concerned with only one chapter (system) of arithmetic. In his work he therefore seeks to explain all the “kinds” of arithmetic in use.

The Islamic world knew three arithmetical systems; finger reckoning, the sexagesimal scale, and Indian arithmetic. Not long after the last was introduced, Greek mathematical writings became accessible and the works of Euclid, Nicomachus, and others were made known. All these elements underwent a slow unification. Abū Mansūr presents them at an intermediary stage in which each system still had its characteristics preserved but was already enriched by concepts or schemes from other systems.

Abū Mansūr conceived of seven system. The first two were the Indian arithmetic of integers and that of fractions. The third was the sexagesimal scale, expressed in Hindu numerals and treated in the Indian way.

The fourth was finger reckoning. Two works on Arabic finger reckoning before the time of Abū Mansūr are extant: the arithmetic of Abu’l-Wafā and that of al-Karajī (known also as al-Karkhī) Both works devote the most space to explaining a cumbersome and complicated fractional system that lacks the idea of the unrestricted common fraction. This system does not appear in the work of Abū Mansūr, who seems to prefer the Indian system. His finger reckoning is confined to concepts lacking in Indian arithmetic, such as shortcuts, and to topics taken from Greek mathematics, such as the summation of finite series. He provides rules for the summation of the general arithmetic, and some special geometric progressions, as well as the sequences r², r³, r⁴ (2r)², (2r— 1)², and polygonal numbers. These rules are expressed in words and assume that the number of terms in each case is ten, a Babylonian practice presented in the works of Diophantus.

Abū Mansūr’s next two systems are the arithmetic of irrational numbers and the properties of numbers. In the first of these. Euclid’s rules of the irrationals in Book X of the Elements are given on a numerical basis. In the second the Pythagorean theory of numbers is presented with an improvement upon Nicomachus: To determine whether n is prime, test it for divisibility by primes Perfect numbers, such as 6, 28, 496, and 8.128, end in 6 or 8; but there is no perfect number between 10⁵ and 10⁶. The first odd abundant number is 945.

This part of Abū Mansūr’s work is ten chapters long, but some folios of the manuscript are missing: only the first three chapters and a few lines of the last are extant. The latter contain an attempt to divide a cube into several cubes by using the relation 3³ + 4³ + 5³ = 6³.

The last of Abū Mansūr’s seven systems, business arithmetic, begins with business problems and ends with two chapters on curiosities that would find a place in any modern book on recreational problems or the modulo principle. One example is given here because it is found in Greek. Indian, and Chinese sources: Your partner thinks of a number not greater than 105. He casts out fives and is left with a: he casts out sevens and is left with b he casts out threes and is left with c. Calculate 21a + 15b + 70c: cast out 105’s, and the residue is the number. The explanation shows that the author was quite familiar with the modulo concept.

Abū Mansūr’s work also seems to solve a problem encountered by historians of medieval mathematics. Latin arithmeticians of the early Renaissance were divided into abacists and algorists. The exact significance of each name was unknown. It has recently been learned that Hindu-Arabic arithmetic required the use of the abacus, thus abacists were those who used the Hindu-Arabic system, and algorists must have adhered to the older system. This agrees with the fact that a work by Prosdocimo de Beldamandi containing an outspoken denunciation of the abacus is called Algorithmus, “Algorisf” and Algorithmus come from the name of al-Khwārizmi the first Muslim to write on Indian arithmetic. His work in Arabic is lost, but we have the Algoritmi de numero indorum. believed to be a translation of it.

But why should those who did not use the abacus be called algorists? This question can be answered as follows: Arabic biographers attribute to al-KhwārizmAbū a book called Kitāb al-jam’ wa’l-tafrīq. now lost. It has been commonly accepted that this was the Arabic title of al-Khwūrizmī#x2019;s work on Indian arithmetic. Abū Mansūr, however, refers to this book in his al-Tukmila., and once he quotes methods from it. These methods follow typical finger-reckoning schemes, which indicates that this book of al-khwal-KhwārizmAbūrizmī’s was of the finger-reckoning type- It seems that those who followed this book of al-Khwārizmī’s were called algorists and those who followed his work on Indian arithmetic were the abacists.

A. S. Saidan