### Abu Al Wafa Al Buzajani (328-388H/940-998CE) أبو الوفاء البوزجاني

A leading astronomer, mathematician, and engineer of the Middle Ages, Abu al-Wafa Muhammed al-Buzajani was a Persian-born scientist who migrated to settle in Baghdad. As the name indicates, Abu al-Wafa hailed from Buzajan, now a deserted land in the vicinity of the small town of Torbat-e-Jam of Khurasan province, where he was born in 328 H/ 940 CE.

Abu al-Wafa obtained his basic education in Mathematics in his hometown, or nearby Nishapur, before leaving for Baghdad. His uncles Abu `Umar al-Mughazili, Abu `Abdullah Muhammad ibn Anbasa and Abu al-Ala’ ibn Karnib were his earliest tutors in mathematics and geometry.

His life coincided with political upheavals in the Islamic world and there occurred a transition of power to a new dynasty that ruled from 940 to 1055 CE, the Buwayhid family. The Buwayhids, during their golden phase, were credited with sponsoring and patronizing scientific projects like building observatories, Bimaristans, irrigation dams, and other such projects. They also encouraged scientific research and patronized many scientists in that period. In fact, this could be the reason behind al-Buzjani’s shifting from his hometown to Baghdad, the seat of the Buwayhid government. He left for Baghdad in 348H/ 959 CE, to find support and sponsorship at the Bayt al-Hikmah (House of Wisdom) of Baghdad

The high point of the dynasty was during the reign of Ad-dud al-Dawlah followed by that of his son Sharaf al-Dawlah. The latter adopted his father’s enthusiasm in supporting scholars of mathematics and astronomy and ordered the building of an observatory in the garden of his palace. Abu al-Wafa worked, along with other scientists, in this observatory which was commissioned in 988 CE. Among the instruments in the observatory was a quadrant over six meters long and a stone sextant of eighteen meters, the first instance in history where a wall quadrant was used for the accurate measurement of the angular distance from the equator on the celestial sphere located midway between the fixed points of the North and South poles. This is believed to be Abu al-Wafa’s own invention. He studied and discussed different movements of the Moon and discovered variations. However, Sharaf al-Dawlah died in the following year and the observatory was closed.

Some time between 961 and 976 CE he wrote Fima yahtaj ilayhi al-kuttab wal-ummal min `ilm al-hisab (The book on what men of knowledge, workers (in the field) and arithmetic require). In the introduction to this book, he wrote that his book comprised all that an experienced or novice, subordinate or an expert in arithmetic needs to know, the art of civil servants, the employment of land taxes and all kinds of business needed in administrations, rations and proportions, multiplication, division, measurements, land taxes, distribution, exchange and all other practices used by various categories of men for doing business and which are useful to them in their daily life.

It is interesting that during this period there were two types of arithmetic books, those using Indian symbols and those of finger-reckoning type. The business community and laymen in general continued to use their finger arithmetic throughout the tenth century. Abu al-Wafa, in spite of his being an expert in Indian numerals, adopted finger arithmetic throughout the teaching material he authored for the business community. In this system, known as Hisab al-Jumal, the numbers were represented by letters in the ancient order of Arabic alphabets, abjad-hawwaz, etc. (See article on Abjad).

The book, one of the few extant treatises for Abu al-Wafa, dwelt into ratios and fractions that involved business dealings and gave good explanations to arithmetical operations employing integers and fractions. A part of the book was devoted to the branch of mensuration in applied geometry in which he described rules for finding areas of surfaces, volumes of solids and length of lines from simple data of lines and angles. The nature of the book, being a manual of applied arithmetical operations, made it delve into describing and solving various problems encountered in public life with regard to business like taxes and their calculation, crops, their exchange and valuation, units of money, payment to soldiers, issue of permits for ships in rivers and merchants on land, as well as other business-related themes. Of special interest in the book is the reference to rules of negative numbers, possibly the first in medieval Arabic mathematics.

An even more impressive display of his arithmetical genius could be seen in one of his works which discussed the theory of numbers. He gave a demonstration of one of his riddles for numbers that produce the same sum whether added to each other or multiplied: 1 + 2 + 3 = 1 * 2 * 3 = 6. The brilliance and elegance of Persian architecture was not devoid of geometrical involvement. The striking beauty and harmony of the varieties of patterns that characterized the interiors as well as the exteriors of domes indicate the involvement of knowledgeable artisans or mathematicians of that time. His other manual Al-Handasa is believed to have been written much later than the earlier manual, although, as the Encyclopedia of Islam suggests, it might have been a collection of his lectures penned down by one of his students. It mentioned the interactions of artists and artisans with mathematicians on topics such as geometric constructions of ornamental patterns and the application of geometry to architectural construction.

The book came in thirteen chapters addressing the design and testing of drafting instruments, the construction of right angles, approximate angle trisections, constructions of parabolas, regular polygons and methods of inscribing them in and circumscribing them about given circles, inscribing of various polygons in given polygons, the division of figures such as plane polygons, and the division of spherical surfaces into regular spherical polygons. It also accommodated, says Suter, a large number of geometrical problems for the fundamental construction of plane geometry to the construction of the corners of a regular polyhedron on the circumscribed sphere.

Another interesting aspect of this particular work of Abu al-Wafa’s is that he tries, where possible, to solve his problems with ruler and compass constructions. When this is not possible, he uses approximate methods. However, there are a whole collection of problems which he solves using a ruler and fixed compass, that is one where the angle between the legs of the compass is fixed, which he believed would give more exact results than can be obtained by changing the compass opening. Abu al-Wafa was given the title Mohandes by the mathematicians, scientists, and artisans of his time, which meant “the Geometer par excellence.” Abu al-Wafa’s contributions in the field of trigonometry were extensive and recognized for the first use of the tan function and for the compilation of tables of sines and tangents at 15’ intervals. He also introduced the sec and cosec and studied the interrelations between the six trigonometric lines associated with an arc. His new methods of calculating sine tables made them accurate to eight decimal tables (converted to decimal notation); Ptolemy’s were only accurate to three places. In the words of EI:

“The chief merit of Abu ‘l-Wafa consists in the further development of trigonometry; it is to him that we owe, in spherical trigonometry, for the right-angled triangle, the substitution, for the perfect quadrilateral with the proposition of Menelaus, of the so called “rule of the four magnitudes” (sine a : sine c = sine A: 1), and the tangent theorem (tan. a : tan. A = sine b : 1); from these formulae he further infers : cos. c = cos. a. cos. b. For the oblique-angled spherical triangle he probably first established the sine proposition (cf. Carra de Vaux, loc. cit., 408-40). We are also indebted to him for the method of calculation of the sine of 30’, the result of which agrees up to 8 decimals with its real value (Woepke, in JA , 1860, 296 ff.).”

There are other works of al-Buzajani which include his Kitab al-Kamil (The Complete Book), a simplified version of Ptolemy’s Almagest. Although there seems to have been little of novel theoretical interest in this work, the observational data in it seem to have been used by many later astronomers. Apart from those above-mentioned, he wrote many other books of which only titles remain, the books are lost. He composed rich commentaries on Euclid, Diophantus, and al-Khawarizmi; however, all of these are not traceable now. His works on the orbit of the moon earned him lasting fame, a crater in the moon being named after him next to many other such craters named after great astronomers, mathematicians, and other scientists.