Kushy?r Ibn Labb
Kushy?r Ibn Labb?n B?shahr?, Abu-’l-?asan, Al-J?l?
(fl.Ca. 1000)
astronomy, trignometry, arithmetic.
Little known about Kushy?r’s life. The word “al-J?l?” added to his name refers to J?l?n, a region of northern Iran south of the Caspian Sea.
The earliest arabic biographers to write about Kushy?r is al-Bayhaq? (d. 1065), who states that Kushy?r lived in Baghdad and died about A. H. 350 (A. D. 961). Later biographers copy al-Bayhaq? and add several attributes to Kushy?r name, including the title “al-kiya,” which seems to mean “master.” But ‘Al? ibn A?mad al-Nasaw?, an arithmetician who flourished after 1029, is said to have been a student of Kushy?r’s. This makes 961 too early; and accordingly, Schoy, Suter, and Brockelmann state that Kushy?r must have flourished between 971 adn 1029. It may be pointed out, however, that Ibn al-Nad?m does not mention Kushy?r,. Ibn al-Nad?m completed the main bulk of his Fihrist about 987 but continued to make additions to it until about 995. It would be rather strange if al-kiya Kushy?r, the prolific writer, had lived in the same city at the same time and remained unnoticed by Ibn al-Nad?m. In his study of Kushy?r’s z?jes, kennedy points out that most of them were probably written after 1000. Accordingly, until further evidence appears, it will be safe to state only that Kushy?r ibn Labb?n florished around A. D. 1000.
The works attributed to kushy?r have survived, but of these only three have received scholarly attention, two z?jes and an arithmetic. The two z?jes are al-J?mi‘, “The comprehensive,” and al-B?ligh, “The Far-reaching.” Each is in four sections: introductory notes, tables, explanations, and proofs. Of the al-B?ligh only the first two sections are extant in the Berlin manuscript. In his “Survey of Islamic Astrrtonomical Tables,” E. S. Kennedy refers to the doubts whether Kushy?r actually wrote two distinct z?jes and gives the impression that al-B?ligh is an abbreviated copy of al-J?mi?.
Kushy?r’s arithmetic is U??l ?is?b al-Hind, “Elements of Hindu Reckoning.” There is a Hebrew commentary to this work written by ‘An?b? in the fifteenth century.
Kushy?r also wrote al-L?mi? f? amthilat al-z?j al-J?mi?, “The Brilliant [Work] on the Examples Pertaining to al-J?mi? z?j”; Kit?b al-As?url?b wa kayfiyyat ‘amalih? wa i?tib?rihi, …, “A Book on the Astrolable and How to Prepare It and Test It …”; Tajr?d U??l Tark?b al-Jur?b, “Extracts of the Principles of Building up Sine Tables”; al-Madkhal [or al-Mujmal] f? ?in?’ at ahk?m al-Nuj?m, “An Introduction [or Summary] of the Rules of Astrology [and Astronomy]”; and Ris?la f? al-Ab??d wa al-Ajr?m, “A Treatise on Distances adn Sizes,” i.e., measuration.
It is believed that Kushy?r did not make any astronomical observations of his own; his z?jes are classified with a few others called “al-Batt?n?’s group.” These take their elements from Muhammad ibn J?bir ibn Sin?n al-Batt?n?’s al-Z?j al-s??bi’?.
Kushy?r is, however, credited with having developed the study of trignometric functions started by Abu’lWaf?? and al-Batt?n?. Abu’l-Waf?? gives sine tables, and al-Batt?n? gives sines and cotangents; but Kushy?r’s z?jes contain sines, cotangents, tangents, and versed, together with tables of differences. In most of these tables the functions are calculated to three sexagesimal places and the angles increase in steps of one degree.
Kushy?r’s unique position in the development of Hindu arithmetic is not yet well understood. The Muslims inherited two arithmetical systems; the sexagesimal system, used mainly by astronomers, and finger reckoning, used by all. Finger reckoning contained no numeration. Numbers were stated in words, and calculations were done mentally. To remember intermediary results, calculators bent their fingers in distinct conventional ways; hence the name finger reckoning. Scribes were able to denote manually numbers from 1 to 9,999, one at a time.
The sexagesimal system used letters of the Arabic alphabet for numeration. Its fraction were always in the scale of sixty, but integers could be in the scale of sixty or of ten. We have arithmetic books that explain the concepts and practices of finger reckoning—most important being the arithmetic written by Abu’l-Waf?? for state astronomers performed their calculations in the scale of sixty before the Indian mwethods began to exert their influence.
Almost every z?j starts by giving arithmetical rules stated rhetorically. Mainly these comprise rules of multiplication and division that may be expressed as
60m? 60n = 60m+n
60m÷ 60n= 60m-n
In books on finger reckoning and on Hindu arithmetic we find statements describing sexagesimal algorisms; these sum to bear Hindu influence. One is thus left with the impression that before they learned Hindu arithmetic, astronomers, like finger reckoners made their calculations mentally, probably depending on finger reckoning as well as sexagesimal calculation. It should be stressed, however, that in Islam the system of finger reckoning used fractions to the scale of sixty and was very rich: it comprised algebra, mensuration, and the elements of trigonometry. it must therefore have been elaborated by the more gifted mathematicians. But whatever the case may be, there remains the question of whether there were any special manipulational methods devised for astronomical calculations.
Abu’l-Waf?? was more of an astronomer than an arithmetician; but since his arithmetic was expressly written for state officials, he may have deliberately avoided bothering his readers with material that he considered too advanced for them.
The importance of Kushy?r’s Us??l H?is?b al-Hind lies in his living written it to introduce the Hindu methods into astronomical calculations. Ab? Han?fa al-D?nawar?, a lawyer, wrote on arithmetic to introduce these methods into business. ?Al? ibn Ahmad al-Nasaw?, known to have been Kushy?r’s student, Ab? ?anif? was lengthy and Kushy?y’s was compact; he said that the former proved to be for astronomers and the latter for businessmen. But al-Nasaw? copied Kushy?y freely in his work and showed no better understanding of the Hindu system.
Kushy?y Us??l is in two sections supplemented by a chapter on the cube root. The first section gives the bare rudiments of Hindu numeration and algorism on the dust board, and in at least one place we find that Kushy?r was not well informed on the new practice. wishing to solve 5625 – 839, like any other computer he puts the array.
5625
839
on the dust board.
Other Arabic writers on Hindu artihmetic (excluding al-Nasaw?) would start by substacting 8 from 6; as this was not possible, they “borrowed” 1 from the 5, broke it down to 10, adn thus subtracted 8 from 16. This process of borrowing and breaking down was not known to Kushy?r (adn his student). He subtracted 8 from 56 complete, obtaining 48. His next step was to subtract 3 from 82.
The second section of Kushy?r’s U??l presents calculations in the scale of sixty, using Hindu numerals and the dust board. Here it seems, although it cannot always be proved decisively schemes alien to with concepts and manipulational schemes alien to both the Hindu system and what is found in books on finger reckoning. These must be schemes in the scale of sixty practised by astronomers.
Briefly Kushy?r states that the scale of sixty is indispensable because it is precise. In the absence of decimal fractions, which were added to the Hindu system by Muslim arithmeticians, Kushy?r’s statement is true. He then shows how to convert decimal integers to this scale; this is necessary for the methods he presents. His multiplication, division, an extraction of roots require the use of a multiplication table extending from 1 X 1 to 60 X 60, expressed in alphabetic numeration and in the sexagesimal scale. With this background he presents homogeneous methods of addition, subtraction, multiplication, division, and extraction of the square root (and, in the supplementary section, the cube root).
Kushy?r certainly worked on the dust board and resorts to erasure and shifting of numbers from place to place; these were the distinguishing feature of the Hindu methods as they came to the Arabic world. But apart from these, he worked out 49°36’ ÷12°25’ as an endless operation of division in decimal fractions would be worked out. He obtained the answer 3°59’ and had 8’25” left as a remainder. He added that the division could be continued if more precise results are desired.
The same applied to roots. Kushy?r found the square root of 45°36’, obtaining the result 6°45’9”59”’, with a remainder. He added that the process could be continue to find the answer to higher degrees of precision.
From this tratment there is only one step to decimal fraction. Al-Uql?dis? mae it again in the fifteenth; but it was left to Stevin to establish it in his La Disme (1585).
BIBLIOGRAPHY
I. Original Works. Kushy?r’s writings are al-B?ligh (Berlin 5751); al-J?mi? (Leiden, Or. 1054), sec. 1 also available as Cairo MS 213; Us? H?is?b al-Hind (Aya Sofya 4857), in M. Levey and M. Petruck, Principles of Hindu Reckoning (Madison, Wis., 1965), pp. 55–83, also edited with ample comparative notes in Arabic by A. S. Saidan in Majallat Ma?had al-Makht??t??t (Cairo, 1967); al-L?mi?fi amthilat al-zij al-j?mi’ (Paris, BN 3487; Cario, MS 138); Tajr?d Us?l Tarkib al-Juy?b (Jarullah 1499/3); al-Madkhal or al-Majmal fi sin??at ahk?m al-Nuj?m (Berlin 5885; Escorial 972; British Museum 415); and Ris?la fi al-ab??d wa al-Ajr?m (Patna).
II. Secondary Literature. See C. Brockelmann, Geschichte der arabischen Literature (Leiden, 1943); E. S. Kennedy, “A Survey of Islamic Astronomical Tables,” in Transactions of the American Philosophical Society, n.s. 46 , pt. 2 (1966); M. Krause, “Stambuler Handschriften islamischer Mathematiker,” in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, abt. B, Studien, 3 (1936). 472–473; C. A. Nallino, AlBattani sive Albattanii opus astronomicum, 3 vols. (Milan, 1899–1907); C. Schoy, “Beiträge zur arabischen Trigonometrie,” in Isis, 5 (1923), 364–399; and H. Suter, “Die Mathematiker und Astronomen der Araber und ihre Werke,” in Abhandlungen zur Geschichte der Mathematik, 10 (1900); and “Nachträge und Berichtigungen zu ’Die Mathematiker und Astronomen . . .,’” ibid., 14 , no. 2 (1902).
A. S. Saidan