Abu al-Wafa Buzjani was a distinguished Muslim astronomer and mathematician, who made important contributions to the development of trigonometry. He worked in a private observatory in Baghdad, where he made observations to determine, among other astronomical parameters, the obliquity of the ecliptic, the length of the seasons, and the latitude of the city.
Background
Abu al-Wafa Buzjani was born in Būzjān or Būzhgān (Khurāsān, Iran) on 10 June 940, in the region of Nīshāpūr, located now in Khurāsān, Iran. He was from an educated and well‐established family.
Abu al-Wafa flourished in an age of great political upheavals. The Būyids (reigned from 945 to 1055), a family originally from the highlands of Daylam in northern Iran, had established a new dynasty that soon extended its rule over Iraq, the heart of the ʿAbbāsid caliphate, reducing the caliph's rule to a mere formality. Under the Būyids, who were great patrons of science and the arts, many scientists and scholars were attracted to Baghdad to enjoy the benefits of the new rulers' patronage. The change in the political climate had brought with it a great cultural revival in the eastern Islamic lands, promoting literary, scientific, and philosophical activities on a large scale.
Education
Abu al-Wafa' Buzjani studied arithmetic under his paternal uncle, Abū ʿAmr Moḡāzelī, and his maternal uncle, Abū Abdallāh Moḥammad Abasa, presumably at Būzǰān.
Career
In 959 at the age of 20, Abu al-Wafa moved to Baghdad, which was then the capital of the Eastern Caliphate, where he soon rose to prominence as a leading astronomer and mathematician at the Būyid court, conducting observations and research in the Bāb al‐Tibn observatory. The decade following 975 seems to have been his most active years in astronomy, during which he conducted most of his observations. Later, to comply with the wishes of the Būyid regent Sharaf al‐Dawla, who was himself a learned man with keen interest in astronomy, Al-Būzjānī became actively involved in the construction of a new observatory in Baghdad. His collaborator was Al-Kūhī, another celebrated astronomer, who excellent in mathematics, physics and in making astronomical instruments. The astronomical work of Al-Būzjānī and his colleagues in Baghdad mark the revival of the “Baghdad school,” a tradition with much vitality in the preceding century.
Abu al-Rayhan al-Bīrūnī, the renowned astronomer and scientist living at that time in Kath (in central Asia), tells us of his correspondence with Al-Būzjānī, who was in Baghdad. In 997, the two astronomers prearranged to make a joint astronomical observation of a lunar eclipse to establish the difference in local time between their respective localities. The result showed a difference of approximately one hour between the two longitudes, very close to present‐day calculations. Al-Biruni made numerous references to Al-Būzjānī's measurements in his various works.
Al-Būzjānī's principal astronomical work, and his sole extant writing on the subject, is Kitāb al‐Majisṭī, which was edited and translated in 2010 by Ali Musa. The book consists of three chapters: trigonometry, application of spherical trigonometry to astronomy, and planetary theory. Although Kitab al‐Majisṭī did not introduce considerable theoretical novelties, it contains observational data that were used by many later astronomers. Its section on trigonometry was a comprehensive study of the subject, introducing proofs in a masterly way for the most important formulas in both plane and spherical trigonometry. Abu al-Wafa’s approach, at least in some instances, bears a striking resemblance to modern presentations.
In this book, Al-Būzjānī introduced for the first time the tangent function and hence facilitated the solutions to problems of the spherical right‐angled triangle in his astronomical calculations. He also devised a new method for constructing the sine tables, which made his tables for sin 30′ more precise than those of his predecessors. This was an important advance, since the precision of astronomical calculations depends upon the precision of the sine tables. The sine table in Al-Būzjānī's Almagest was compiled at 15' intervals and given to four sexagesimal places. In the sixth chapter of the book, Al-Būzjānī defines the terms tangent, cotangent, sine, sine of the complement (cosine), secant and cosecant, establishing all the elementary relations between them.
In mathematics, Abu al-Wafa’s contributions cover both theoretical and practical aspects of this science. His practical textbook on geometry, A Book on Those Geometric Constructions Which Are Necessary for a Craftsman (Kitab fima yahtaju ilayhi al-sani’ min ‘ilm al-handasa), is unparalleled among the geometrical works of its kind written in the Islamic world. He wrote also a practical textbook on arithmetic, the Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen (Kitab fima yahtaju ilayhi al-‘ummal wa-‘l-kuttab min ‘ilm al-hisab).
On the basis of works attributed to him, Al-Būzjānī seems to have been a prolific scholar. He is said to have written 22 books and treatises. These include works on astronomy, arithmetic, and geometry, as well as translations and commentaries on the algebraic works of past masters like Diophantus and Al-Khwārizmī, and a commentary on Euclid's Elements. Of all these works, however, we know of only eight which have survived. Of his astronomical works, references were made to a Zīj al‐wāḍiḥ, an influential work that is no longer extant.
Historical evidence, as well as the judgments of Būzjānī's colleagues and generations of scholars who came after him, all attest to the fact that he was one of the greatest astronomers of his age. He was also said to have been a man with great moral virtues who dedicated his life to astronomy and mathematics. His endeavors in the domain of science did not die with him. In fact, the data he had gathered from his observations were used by astronomers centuries after him. Furthermore, the science of trigonometry as it is today is much indebted to him for his work.
Religion
In his religious affiliation Abu al-Wafa' Buzjani was a Muslim.
Views
Abu al-Wafa al-Buzjanî 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-Khwarizm. None of these commentaries has yet been found.
Abu’l-Wafa’s textbook on practical arithmetic, Kitab fi ma yahtaj ilayh al-kuttab wa’l-cummal min cilm al-hisab (“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 (manazil), each of which has seven chapters (abwab). 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.
Abu’l-Wafa 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.” Abu’l-Wafa 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 qumbers and fractions.
Abu’l Wafa’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, Abu’l-Wafa 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 > 1, was uncommon outside the circle of specialists. Merchants and other businessmen had long used as their basic fractions - called ras (“principal fractions”) by Abu’l-Wala - 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.
Abu’l-Wafa 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 (ksaya).
Some historians, such as M. Cantor and H. Zeuthen, explain the lack of positional numeration and “Indian” numerals in Abu’l-Wafaz’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 Abu’l-Wafas, the above computation of fractions, for example, is found in a book by al-KarajI at the beginning of the eleventh century and in works by other authors.
Abu’l-Wafa 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 7. 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 Abu’l-Wafa is Kitdb fl mayahtaj ilayh al-sani min al-acmal 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 Abu’l-Wafas 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 constructions (the division of a segment into equal parts, the construction of a tangent to a circle from a point on or outside the circle, to the construction of regular and semiregular 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, intercalation is used (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 a compass with an invariable opening deserves mention. Such constructions are found in the writings of the ancient Indians and Greeks, but Abu'l-Wafa 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 obtained 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 constructions.
Also in this work by Abu’l-Wafa 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, Abu’l-Wafa simultaneously proved the inaccuracy of some methods used by “artisans.”
Abu’l-Wafa large astronomical work, al-majistj, or Kitab al-kamil (“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 Zijal-Wadih, based on observations that he and his colleagues conducted. The Zij seems not to be extant. Abu’l-Wafa 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. E. S. Kennedy established that the data from Abu’l-Wafa observations were used by many later astronomers.
Abu’l-Wafa`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.
In comparison, Ptolemy’s method of interpolation, which was used before AbuT-Wafa, showed error in the third place. If one expresses Abu’l-Wafa`s approximation in decimal fractions and lets r = 1 (which he did not do), then sin 30' = 0.0087265373 is obtained instead of 0.0087265355 - that is, the result is correct to 10-8. Abu’l-Wafa also compiled tables for tangent and cotangent.
In spherical trigonometry before Abu’l-Wafa 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. Abu’l-Wafa 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-Blrflni. One of the first proofs of the general theorem of sines applied to the solution of oblique triangles also was originated by Abu’l-Wafal. In Arabic literature this theorem was called “theorem which makes superfluous” the study of complete quadrilaterals and Menelaus’ theorem.
