Ab?
Ab?’l-Waf?? Al-B?zj?n?, Mu?ammad Ibn Mu?ammad Ibn Ya?y? Ibn Ism???l Ibn Al-?Abb?s
(b. B?zj?n [now in Iran], 10 June 940; d. Baghdad [now in Iraq], 997 or July 998)
mathematics, astronomy.
Ab?’l-Waf?? was apparently of Persian descent. In 959 he moved to Baghdad, which was then the capital of the Eastern Caliphate. There he became the last great representative of the mathematics-astronomy school that arose around the beginning of the ninth century, shortly after the founding of Baghdad. With his colleagues, Ab?’l-Waf?? conducted astronomical observations at the Baghdad observatory. He continued the tradition of his predecessors, combining original scientific work with commentary on the classics—the works of Euclid and Diophantus. He also wrote a commentary to the algebra of al-Khw?rizm?. None of these commentaries has yet been found.
Ab?’l-Waf??’s textbook on practical arithmetic, Kit?b f? m? ya?taj ilayh al-kutt?b wa’l-?umm?l min ?ilm al-?is?b (“Book on What Is Necessary From the Science of Arithmetic for Scribes and Businessmen”), written between 961 and 976, enjoyed widespread fame. It consists of seven sections (man?zil), each of which has seven chapters (abw?b). The first three sections are purely mathematical (ratio, multiplication and division, estimation of areas); the last four contain the solutions of practical problems concerning payment for work, construction estimates, the exchange and sale of various grains, etc.
Ab?’l-Waf?? systematically sets forth the methods of calculation used in the Arabic East by merchants, by clerks in the departments of finance, and by land surveyors in their daily work; he also introduces refinements of commonly used methods, criticizing some for being incorrect. For example, after indicating that surveyors found the area of all sorts of quadrangles by multiplying half the sums of the opposite sides, he remarks, “This is also an obvious mistake and clearly incorrect and rarely corresponds to the truth.” Ab?’l-Waf?? does not introduce the proofs here “in order not to lengthen the book or to hamper comprehension,” but in a series of examples he defines basic concepts and terms, and also defines the operations of multiplication and division of both whole numbers and fractions.
Ab?’l-Waf??’s book indicates that the Indian decimal positional system of numeration with the use of numerals—which Baghdad scholars, acquainted with it by the eighth century, were quick to appreciate—did not find application in business circles and among the population of the Eastern Caliphate for a long time. Considering the habits of the readers for whom the textbook was written, Ab?’l-Waf?? completely avoided the use of numerals. All numbers and computations, often quite complex, he described only with words.
The calculation of fractions is quite distinctive. Operation with common fractions of the type m/n, where m, n are whole numbers and m > l, was uncommon outside the circle of specialists. Merchants and other businessmen had long used as their basic fractions—called ra’s (“principal fractions”) by Ab?’l-Waf??—those parts of a unit from 1/2 to 1/10, and a small number of murakkab (“compound fractions”) of the type m/n, with numerators, m, from 2 to 9 and denominators, n, from 3 to 10, with the fraction 2/3 occupying a privileged position. The distinction of principal fractions was connected with peculiarities in the formation of numerical adjectives in the Arabic language of that time. All other fractions m/n were represented as sums and products of basic fractions; businessmen preferred to express the “compound” fractions, other than 2/3, with the help of principal fractions, in the following manner;
Any fraction m/n, the denominator of which is a product of the sort 2P 3Q 5R 7S, can be expanded into basic fractions in the above form. In the first section of his book, Ab?’l-Waf?? explains in detail how to produce such expansions with the aid of special rules and auxiliary tables. Important roles in this operation are played by the expansion of fractions of the type a/60 and the preliminary representations of the given fraction m/n in the form m · 60/n ÷ 60 (see below). Since usually for one and the same fraction one can obtain several different expansions into sums and products of basic fractions, Ab?’l-Waf?? explains which expansions are more generally used or, as he wrote, more “beautiful.”
If the denominator of a fraction (after cancellation of the fraction) contains prime factors that are more than seven, it is impossible to obtain a finite expansion into basic fractions. In this case approximate expansions of the type 3/17 ? (3 + 1) ÷ (17 + 1) = 2/9 or 3/17 ? 31/2 ÷ 171/2 = 1/5—or still better, 3/17 ? 31/7 ÷ 171/7 = 1/6 + 1/6 · 1/10—were used.
Instead of such a method, which required the skillful selection of a number to be added to the numerator and denominator of a given fraction, Ab?’l-Waf?? recommended the regular method, which enables one to obtain a good approximation with reasonable speed. This method is clear from the expansion
Analogously, one can obtain
or
The error of this last result as Ab?’l-Waf?? demonstrates, equals
The calculation described somewhat resembles the Egyption method, but, in contrast with that, in (1) is limited to those parts of a unity 1/q, for which 2 ? q ? 10; (2) uses products of the fractions 1/q1 · 1/q2 and 2/3 · 1/q; and (3) does not renounce the use of compound fractions m/n, 1 < m < n ? 10. Opinions differ regarding the origin of such a calculation; many think that its core derives from ancient Egypt; M. I. Medovoy suggests that it arose independently among the peoples living within the territory of the Eastern Caliphate.
In the second section is a description of operations with whole numbers and fractions, the mechanics of the operations with fractions being closely connected with their expansions into basic fractions. In this sections there is the only instance of the use of negative numbers in Arabic literature. Ab?’l-Waf?? verbally explains the rule of multiplication of numbers with the same ten’s digit:
(10a + b) (10a + c) = [10a + b – {10(a + 1) – (10a + c)}]10(a + 1) + [10(a + 1) – (10a + b)] · [10(a + 1) – (10a + c)].
He then applies it where the tens digit is zero and b = 3 and c = 5. In this case the rule gives.
3 · 5 = [3 – (10 – 5)] · 10 + [10 – 3] · [10 – 5] = (– 2) · 10 + 35 = 35 – 20.
Ab?’l-Waf?? termed the result of the subtraction of the number 10–5 from 3 a “debt [dayn] of 2.” This probably reflects the influence of Indian mathematics, in which negative numbers were also interpreted as a debt (k?aya).
Some historians such as M. Cantor and H. Zeuthen, explain the lack of positional numberation and “Indian” numerals in Ab?l-Waf??’s textbook, as well as in many other Arabic arithmetic courses, by stating that two opposing schools existed among Arabic mathematicians: one followed Greek models the other Indian models; M.I. Medovoy, however, shows that such a hypothesis is not supported by fact. It is more probable that the use of the Positional “Indian” arithmetic simply spread very slowly among businessmen and the general population of the Arabic East, Who for a long time preferred the customary methods of verbal expression of whole numbers and fractions and of operations dealing with them. Many authors considered the needs of these people; and, after Ab?’l-Waf??, the above computation of fractions, for example, is found in a book by al-Karaj? at the beginning of the eleventh century and in works by other authors.
In the third section Ab?’l-Waf?? gives rules for the measurement of more common planar and three-dimensional figures—from triangles various types of quadrangles. regular polygons, and a circle and its parts, to a sphere and sectors of a sphere, inclusive. There is table of Chords corresponding to the the arcs of a semicircle of radius 7, which consists of m/22 of the semicircumference (m = 1,2,..., 22), and the expression for the diameter, d, of the a circle sup–rscribed around a regular n–sided polygon with side a:
Ab?’l-Waf?? thought this rule was obtained from India; it is correct for n = 3, 4, 6, and for other values of n gives a good approximation, especially for small n. At the end of the third section, problems involving the determination of the distance to inaccessible objects and their heights are solved on the basis of similar triangles.
Another practical textbook by Ab?’Waf?? is Kit?b f? m? ya?taj ilayh al-??ni? min al-a?m?l al-handasiyya (“Book on what is necessary from Geometric Construction for the Artisan”), written after 990. Many of the two-dimensional and three-dimensional constructions set forth by Ab?’l-Waf?? were borrowed mostly from the writings of Euclid, Archimedes, Hero of Alexandria, Theodosius, and Pappus. Some of the examples, however, are original. The range of problems is very wide, from the simplest planar constructiontions (the division of a segment into equal parts, the constructions of tangent to a circle from a point on or outside the circle, etc.) to the construction of regular and serniregular polyhedrons inscribed in a given sphere. Most of the constructions can be drawn with a compass and straightedge. In several instances, when these means are insufficient, intercalations is use (for the trisection of an angle or the duplication of a cube)or only an approximate construction is given for the side of a regular heptagon inscribed in a given circle, using half of one side of an equilateral triangle inscribed in the same circle, the error is very small).
A group of problems that are solved using a straightedge and compass with an invariable opening deserves mention. Such constructions are found in the writings of the ancient Indians and Greeks, but Ab’’Waf?’ was the first to solve a large number of problems using a compass with an invariable opening. Interest in these constructions was probably aroused by the fact that in practice they give more exact results than can be obtianed by changing the compass opening. These constructions were widely circulated in Renaissance Europe; and Lorenzo Mascheroni, Jean Victor Poncelet, and Jakob Steiner developed the general theory of these and analogous construtions.
Also in this work by Ab?’l-Waf?? are problems concerning the division of a figure into parts that satisfy certain conditions, and problems on the transformation of squares (for example, the construction of a square whose area is equal to the sum of the areas of three given squares). In proposing his original and elegant constructions, Ab?’l-Waf?? simultaneously proved the inaccuracy of some methods used by “artisans.”
Ab?’l-Waf??s large astronomical work, al-majis??, or Kit?b al-k?mil (“Complete Book”), closely follows Ptolemy’s Almagest. It is possible that this work, available only in part, is the same as, or is included in, his Z?j al-W??i?, based on observations that he and his colleagues conducted. The Z?j seems not to be extant. Ab?’l-Waf?? apparently did not introduce anything essentially new into theoretical astronomy. In particular, there is no basis for crediting him with the discovery of the so-called variation of the moon (this was proved by Carra de Vaux, in opposition to the opinion expressed by L.A. Sédillot).E. S. Kennedy established that the data from Ab?’l-Waf??s observations were used by many later astronomers.
Ab?’l-Waf??s achievements in the development of trigonometry, specifically in the improvement of tables and in the means of solving problems of spherical trigonometry, are undoubted. For the tabulation of new sine tables he computed sin 30 more precisely, applying his own method of interpretation. This method, based on one theorem of Theon of Alexandria, gives an approximation that can be stated in modern terms by the inequalities
The values sin 15°/32 and sin 18°/32 are found by using the known values of sin 60° and sin 72°, respectively, with the aid of rational operations and the extraction of a square root, which is needed for the calculation of the sine of half a given angle; the value sin 12°/32 is found as the sine of the difference 72°/32–60°/32. Setting sin 30’ equal to half the sum of the quantities bounding it above and below, with the radius of the circle equal to 60, Abül-Waf?? found, in sexagesimal fractions, sin 30? = 31I 24II 55III 54IV 55V. This value is correct to the fourth place, the value correct to five places being sin 30’ = 31I 24II 55III 54IV 0V.
In comparison, Ptolemy’s method of interpolation, which was used before Ab?’l-Waf??, showed error in the third place. If one expresses Ab?’l-Waf??’s approximation in decimal fractions and lets r = 1 (which he did not do), then sin 30’ = 0.0087265373 is obtained instead of 0.008725355–that is, the result is correct to 10–8;. Abu?’Wafa? also compiled tables for tangent and cotangent.
In spherical trigonometry before Abu’l-Waf??, the basic means of solving triangles was Menelaus’ theorem on complete quadrilaterals, which in Arabic literature is called the “rule of six quantities.” The application of this theorem in various cases is quite cumbersome. Ab?’l-Waf?? enriched the apparatus of spherical trigonometry, simplifying the solution of its problems. He applied the theorem of tangents to the solution of spherical right triangles, priority in the proof of which was later ascribed to him by al-B?r?n?. One of the first proofs of the general theorem of sines applied to the solution of oblique triangles also was originated by Ab?’l-Waf??. In Arabic literature this theorem was called “theorem which makes superfluous” the study of complete quadrilaterals and Menelaus’ theorem. To honor Ab?’l-Waf??, a crater on the moon was named after him.
BIBLIOGRAPHY
I. Original Works. The text on practical arithmetic has never been published in any modern language; however, discussions of it may be found in Woepeke, Luckey, and Medovoy (see below). Manuscripts of this work are preserved at the library of the University of Leiden(993) and the National Library, Cairo (1V, 185). Besides these, there exist the manuscript of a work containing the fundamental definitions of theoretical arithmetic: “Risãla fi’l-aritmatiqi,” at the Institute of Oriental Studies of the Academy of Sciences of the Uzbek S.S.R. in Tashkent (4750/8), which is described by G. P. Matvievskaja (see below); an unstudied arithmetical manuscript at the Escorial (Casiri, 933); and an unstudied arithmetical manuscript at the Library Raza, Rampur (I,414). The text on geometric constructions has been studied in a Persian variant (Paris, Bibliothéque Nationale, pers, anc., 169) by Woepcke: Suter has studied the Milan manuscript (Biblioteca Ambrosiana, arab. 68); and a Russian translation by Krasnova of the Istanbul manuscript (Aya Sofya, 2753) has appeared. Eleven of the thirteen chapters of the latter are extant. The manuscript of the al-Majis??, only part of which has survived, is in Paris (Bibliothéque Nationale, ar, 2497) and has been studied by Carra de Vaux (see below). MS Istanbul, Carullah, 1479, is unstudied.
II.Secondary Literature. General works concerning Ab?l-Waf?? are A. von Braunmühl, Vorlesungen über Geschichte der arabischen Litteratur, 1, 2nd ed. (Leiden, 1943), 255; Supp. I (Leiden, 1937), p. 400; M. Cantor, M. Cantor, Vorlesungen über Geschichte der Mathematik, 2nd ed., I (Leipzig, 1894), 698–704, Index; Ibn al-Nad?m (Aûl-Far?j Mu?ammad Ibn Is??q), Kitãb al-Fihrist, G, Flügel, Y.Rödiger, and A. Müller, eds., I (Leipzig, 1871), 266, 283; H. Suter’s translation of the Fihrist, “Das Mathematikerverzeichnis im Fihrist des Ibn Ab? Ya’k?b al-Nad?m,” in Abhandlungen zur Geschichte der mathematischen Wissenschaften, 6 (1892), 39; A. Youschkevitch, Geschichte der Mathematik fin Mittelalter (Leipzig, 1964), Index; G. Sarton, Introduction to the History of Science, I 666–667; H. Suter. The Encyclopaedia of Islam, new ed., I (Leiden–London, 1954), 159, and Die Mathematiker und Astronomen der Araber (Leipzig, 1900–1902), 71–72; Supp., 166–167; and Joh. Tropfke, Geschichte der Elementar Mathematik, 2nd ed., VII (Leipzig, 1921–1924), Index.
The first attention to Ab?l Waf??s work was F. Woepcke’s “Analyse et extraits d’un recueil de constructions géométriques par Aboûl Wefâ,” in Journal asiatique, 5th ser., 5 (1855), 218–256, 309–359, which deals with the Paris (Persian) manuscript. For an analysis of Abu’l-Wafã”s practical arithmetic, see P. Luckey. Die Rechenkunst bei Gamšid b. Mas’?d al-K?ši mit Rückblicken auf die ältere Geschichte des Rechnens (Wiesbaden, 1951). There are two detailed investigations of the arithmetic text by M. I. Medovoy: “Ob odnom sluchae primenenija otritsatel’nykh chisel u Abu-l-Vafy” (“On One Case of the Use of Negative Numbers by Ab?’l-Waf??”), in Istoriko-matematicheskie issledovanija (“Studies in the History of Mathematics”), 11 (1958), 593–598, and “Ob arifmeticheskom traktate Abu-l-Vafy” (“On the Arithmetic Treatise of Ab?’l-Wafa”), ibid., 13 (1960), 253–324; both articles constitute the first detailed investigation of this work. On the Tashkent manuscript, see G. P. Matvievskaja, O matematicheskikh mkopissiakh iz sobranija instituta vostokovedenija AN Uz. S.S.R. (“On the Mathematical Manuscripts in the Collection of the institute of Oriental Studies of the Academy of Sciences of the Uzbek S.S.R.”), Publishing House of the Academy of Sciences of the Uzbek S.S.R. Physical and Mathematical Sciences Series, pt. 9 (1965), no. 3, and Uchenije o chisle na siednevekovom Vostoke (“Number Theory in the Orient During the Middle Ages”): (Tashkent, 1967). An exposition of the Milan geometric manuscript is H. Suter, “Das Buch der geometrischen Konstruktionen des Ab?’l Wefa,” in Abhandlungen zur Geschichte der Naturwissenschaften und Medizin (1922), pp. 94–109. A Russian translation of the Istanbul geometric manuscript, with commentary and notes, has been done by S. A. Krasnova: “Ab?-l-Vafa al-Buzdzhani, Kniga o tom, chto neobkhodimo remeslenniku iz geometricheskikh postroenij” (“Ab?’l-Waf?? al-B?zj?n?, ’Geometrical Constructions for the Artisan”), in Fiziko-matematicheskie naukty stranakh vostoka (“Physics and Mathematics in the Orient”), I (IV) (Moscow, 1966), 42–140.
The astronomical and trigonometrical works of Ab?’lWafa’ are discussed in Carra de Vaux,’L’Almageste d’Ab?–l–Wéf? Almageste d’Ab?-l-Wéf? al-B?zj?n?, in Journal asiatique, 8th ser., 19 (1892), 408–471; E. S. Kennedy, “A Survey of Islamic Astronomical Tables,” in Transactions of the American Philosophical Society, n.s. 46 (1956), 2; and F. Woepcke, “Sur une mesure de la circonférence du cercle due aux astronomes arabes et fondée sur un calcul d’Aboûl Wefâ,” in Journal asiatique, 5th ser., 15 (1860), 281–320.
A. P. Youschkevitch