Development of Algebra during the Middle Ages

Overview

During the Middle Ages, while the intellectuals of Christian Europe concerned themselves with theology and the common people with subsistence agriculture, a vibrant scientific and mathematical culture developed in the Islamic world. Among its achievements was the development of algebra, which would be reintroduced into Western mathematics through the Latin translation of a book, the al-jabr, by the ninth-century Persian astronomer and mathematician al-Khwarizmi.

Background

In its modern sense, algebra is the branch of mathematics concerned with finding the values of unknown quantities defined by the equations that they satisfy. Problems of an algebraic type are recognizable in the surviving mathematical writings of the Egyptians and Babylonians. Ancient Greek mathematics included the development of some algebraic concepts, but mainly in connection with geometry. The Greek mathematicians and philosophers were uncomfortable with the existence of irrational numbers, numbers such as the square root of two; irrational numbers cannot be expressed as the ratio of two whole numbers and are almost unavoidable in algebraic problems. One of the later Greeks to write on algebraic topics was Diophantus (c. a.d. 210-c. 290), who was affiliated with the famous library at Alexandria about the year 250. Diophantus was the first to use symbols to introduce unknown quantities that then allowed equations to be written.

By the time of the fall of the Roman Empire in the fifth century a.d., the study of mathematics had made considerable progress in both China and India. In India it was put to use in the service of astronomy and Hindu religious practices. A number of Indian scholars, notably the astronomer and mathematician Brahmagupta (598-c. 665) made contributions to algebra and arithmetic in the treatment of astronomical problems. Most of these are known from the Brahamasphuta Siddhanta, which includes a discussion of algebraic equations that goes beyond the work of Diophantus, and a treatment of arithmetic using the system of ten numerals, including zero, that is essentially that used today.

In 632 the prophet Muhammad established an Islamic state centered in Mecca. Following his death in the same year, his followers began to conquer lands to the east and west. By 750 Arab armies had established control of a region extending from southern Spain through North Africa, Asia Minor, and into India. Initially hostile to other cultures, the Islamic rulers soon began to welcome the scholars of many different lands within their realm. By 766 an Arabic translation of a major Hindu mathematical work, most likely the Siddhanta of Brahmagupta, had been prepared. At Baghdad, the Caliph al-Ma'mun (786-833) established a "House of Wisdom" modeled on the earlier Greek academy at Alexandria. Al-Ma'mun encouraged the translation of the mathematical works in other languages into Arabic. One of society's members would be the mathematician and astronomer al-Khwarizmi (c. 780-c. 850).

Our word "algebra" comes from al-Khwarizmi's book Kitab al-jabr wa al-muqabala, or The Compendious Book on Calculation by Completing and Balancing. In this book (referred to as the al-jabr) al-Khwarizmi described the art of finding the value of an unknown quantity in equations by rearranging terms. The book had three parts, only one of them dealing with algebra as we know it, the remainder dealing with measurement and with some legal questions concerning inheritances. Most remarkably, the discussion of algebra is presented entirely without symbols; even the numbers are spelled out in words. Thus the quantity (x/3 + 1) would be described by al-Khwarizmi as "a thing, take a third of it and add a unit."

Although al-Khwarizmi, unlike the ancient Greeks, was untroubled by irrational numbers, he was unwilling to accord negative numbers the same status as positive ones. Thus he stated that there are six fundamental forms of equations involving powers no greater than the square of a single unknown quantity. The modern student of algebra would recognize each of these as a special case of the general quadratic equation, ax² + bx + c = 0, with a, b, or c being possibly zero or negative. Al-Khwarizmi's fundamental forms all have positive and nonzero coefficients.

For each of the fundamental forms, al-Khwarizmi explained how a solution may be obtained by a sequence of mathematical operations. He gave rules for solving each of them, using the ordinary arithmetic operations of addition, subtraction, multiplication, and division, plus the taking of square roots. The "al-jabr" of the book's title actually refers to the process of "restoration" or "completion" by which negative quantities appearing in a problem are eliminated to obtain one of the standard forms. "Al-muqabala" refers to the combining of terms involving the same power of the unknown, again to obtain one of the six standard forms.

The al-jabr suggests that al-Khwarizmi was familiar with both the Hindu and Greek traditions in mathematics. Hindu influence is even more apparent in his astronomical work, but the symbol-free presentation of the al-jabr is also more consistent with Sanskrit mathematical texts than with Greek. On the other hand, al-Khwarizmi's use of geometrical figures to illustrate equations suggests a familiarity with Euclid's Elements. While the al-jabr included no symbols, the Hindu system of numerals would be described by him in a second book now known only in Latin translation as Algoritmi de numero Indorum. It is from the first word of this title, meaning "al-Khwarizmi's," that we obtain the modern word "algorithm," meaning a systematic method of obtaining a mathematical result.

Among other Persian scholars to be concerned with algebra was the Persian poet and astronomer Abu'l-Fath Umar ibn Ibrahim al-Khayyami (1048-1131), better known as Omar Khayyam. Khayyam's poetry was translated into English in the nineteenth century, but his mathematical work remained unknown in the West until the 1930s. By 1079 Khayyam had written a manuscript on cubic equations that gave the solution of every type of cubic equations with positive real number solutions. He also discovered a relation between the coefficients generated by the expansion of the binomial (a + b)ⁿ that would be independently discovered by the French mathematician Blaise Pascal (1623-1662) six centuries later.

Impact

The survival and circulation of manuscripts before the invention of the printing press is an uncertain matter. Al-Khwarizmi's al-jabr is in many respects less advanced than earlier works. It is possible that the practical and elementary nature of the presentation was instrumental in its survival while other texts have been lost. Scholars thus are uncertain about how much of the al-jabr was original and how much may have been taken from earlier Arab or Hindu sources. Nonetheless, it is through this book that the study of algebra was reintroduced into Europe during the Renaissance. The al-jabr was translated into Latin in 1145 as the Liber algebrae et almucabala by Robert of Chester (fl. c. 1141-1150), an English scholar living in Islamic Spain.

By the time of the Renaissance, mathematics in Christian western Europe was in a far less advanced state than in the Islamic world. The Byzantine Empire had, however, provided a refuge for Greek-speaking scholars. In 1543, after Turkish forces overran its capital, Constantinople, Byzantine scholars began to seek refuge in Italy, where rich and powerful families like the Medici were collecting manuscripts and providing financial support for scholars. The appearance of Johannes Gutenberg's (1390?-1468) printing press would make mathematical ideas far more widely available. Over 200 new books on mathematics appeared in Italy before 1500.

In 1545 a book entitled Ars Magna or The Great Art by the Italian mathematician Girolamo Cardano (1501-1576) appeared. This work incorporated significant new results, including the solution of the cubic and quartic equations. Cardano used letters to represent known quantities, but unlike Diophantus he stopped short of using other letters for the unknown quantities. The next major step forward in algebra occurred with the work of French lawyer and writer François Viète (1540-1620), who introduced the modern practice of using letters to represent the unknown as well as known quantities. With Viète's new notation, it became easier to think of solving an algebraic equation as finding the values of x for which a definite function of the variable x would equal zero. This set the stage for the study of functions themselves and the study of transformations of functions caused by introducing new variables, ideas important in modern algebra, trigonometry, and calculus.

Algorithms to obtain various results have been known, of course, throughout the history of mathematics. One of the most famous is the algorithm attributed to the Greek mathematician Euclid (fl. c. 300 b.c.) for finding the greatest common factor of two integers. The various recipes given by al-Khwarizmi for transforming equations into one of the six standard forms and for solving each of them are also examples of algorithms. The notion of algorithm would gain increased attention in the twentieth century. Algorithms are, after all, rules for the manipulation of symbols, and nineteenth-century mathematics had become increasingly concerned with the formalization of mathematics—that is, translating the statements of mathematics into purely symbolic form. In 1900 the great German mathematician David Hilbert (1862-1943) proposed that an algorithm might be found that could produce a solution to any mathematical problem expressed as a string of symbols. A young British mathematician, Alan Turing (1912-1954), was prompted by this proposal to examine the notion of algorithm, particularly in arithmetic, far more closely, and ultimately to prove that no such algorithm could exist. In his work, however, Turing first described the operation of a simple symbol processing machine, now called a "Turing machine," that could be built from electronic components and that has evolved into the digital computer of the present day.

DONALD R. FRANCESCHETTI