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The twenty-first-century approach to the history of mathematics looks beyond the once-traditional Eurocentric views that credited the Greeks for the most advanced developments; it now embraces a global view of mathematical accomplishment through the ages and crosscultural exchanges in the field.

The traditional history of mathematics took shape at the turn of the twentieth century, reflecting the European influence of that era. According to this traditional history the first mathematics worthy of the name arose in ancient Greece and led directly, but for the Dark Ages, toward the modern mathematics of Western universities today. Since the mid-twentieth century historians of mathematics have broadened their subject’s horizons. New historical evidence and changing attitudes toward other cultures have made this history of mathematics more complex and more global. Equally important have been changes in what is considered mathematics. The key to the study of a truly global history of mathematics has been flexibility in the definition of mathematics itself. The term ethnomathematics refers to the mathematical practices of a particular society and includes mathematical processes beyond the narrow academic sense, in recreation, art, numeration, and other fields. A more conceptually inclusive definition of mathematics has led directly to a more geographically inclusive history of mathematics. We can see the greatest diversity in mathematical practices by comparing counting systems around the world, such as the Incan quipu, a knot-tying system for recording numbers.

## Toward a Hellenistic Synthesis

Hundreds of well-preserved clay tablets (most dating from c. 1700 BCE) disclose the sexigesimal (base sixty) numeration system of ancient Mesopotamia, a system sufficient for the development of an ancient algebra. The sources for early Egyptian mathematics are far fewer. The earliest, the Ahmes Papyrus, is a copy of an earlier version from about 1900 BCE. In addition to developing an arithmetic distinctive for its reliance on unit fractions (1?4, 1?3, 1?2, etc.), Egyptians elaborated a practical computational geometry with methods for calculating the volume of a truncated pyramid and approximating the area of a circle.

Only recently have historians begun favorably to reevaluate the achievements of Mesopotamia and Egypt because none of their earliest mathematics adopted the deductive and theoretical approach of the Greeks—an approach that has become enshrined in modern definitions of mathematics. The Greeks themselves, however, explicitly acknowledged their debt to the Egyptians. Moreover, after the conquests of Alexander of Macedon (Alexander the Great, 356–323 BCE) spread Greek thought throughout southwestern Eurasia, a shared Hellenistic culture united the mathematicians of the new world order, and the great Greek geometer Euclid (early third century BCE) taught in the Egyptian capital Alexandria. The Hellenistic world witnessed an extraordinary synthesis of Greek deductive geometry with the algebraic and empirical mathematics of neighboring peoples.

## Indian Mathematics

The earliest Indian mathematics was expressly religious in purpose. The geometry developed in the Sulbasutras (technical appendices to Hindu sacred writings, the Vedas, c. 3000 BCE) adopted sophisticated techniques, including the Pythagorean theorem, for the proper orientation and construction of effective sacrificial altars. A hymn from the Atharavaveda (the name translates to Sacred Knowledge of Magical Forms) describes a complex design of overlapping triangles for meditation.

Possibly during the Vedic period, and certainly by the seventh century CE (perhaps inspired by the counting boards brought by Chinese travelers), a decimal place-value system arose that in principle is like our own: distinct numerals represented the numbers 0 through 9, and each instance of a numeral would be associated with a decimal scale based on its position in the number. (The first dated inscription of such a system, however, comes from Cambodia in 683.) Most early mathematical knowledge was preserved orally in verse.

The decline in Vedic sacrifices after about 500 BCE shifted the focus of Indian mathematics. The developing Buddhist and Jaina traditions involved large numbers. (The religion of Jainism, originating in sixth-century-BCE India, taught liberation of the soul.) The collection of Buddhist texts Samyutta Nikaya (The Grouped Discourses), for example, defines a kalpa as a unit of time longer than that needed to erode a rock of one cubic mile by wiping it with silk once a century. The Jainas worked with series and permutations, and they theorized that not all infinite numbers are equal, an insight that would elude Western mathematics until the nineteenth century. Only with the practical, commercial algebraic examples of the Bakhshali manuscript, composed in the first centuries CE, did Indian mathematics lose its primarily religious applications.

Later Indian mathematics would achieve breakthroughs in indeterminate equations, as well as in trigonometry and its application to astronomy. Algebraic work used Sanskrit letters abbreviating the names of colors to stand for unknown qualities, like the modern x and y, and until the modern period the use of letters of the alphabet to represent numerals would reinforce the poetry of Indian mathematics. The research on power series (series—sum of a sequence of terms—in which the terms follow a certain pattern and are infinite) for pi and trigonometric functions by fifteenth- and sixteenth-century mathematicians in Kerala in India anticipated many of the results achieved by the English physicist and mathematician Isaac Newton and the German philosopher and mathematician G. W. Leibniz—independently, or perhaps through the Portuguese—in the seventeenth century. Proofs and derivations of formulas were quite rare throughout the Indian mathematical tradition.

## Chinese Mathematics

The first existing numerals appeared on Chinese Shang dynasty (1766–1045 BCE) tortoise shells used for divination, with symbols to indicate 1 through 10, 20s, 100s, and 1000s. The oldest treatise is the Zhoubi Suanjing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) (500– 200 BCE), an archaic astronomy text that contains a demonstration of the Pythagorean theorem. In China counting rods were used for elementary arithmetical operations at least by the Qin dynasty (221–206 BCE). China also boasts the earliest known magic squares, which are matrixes of numbers in which every row, column, and major diagonal produces the same sum. Interest in magic squares as protective talismans encouraged their spread to Mongolia and Tibet, although they were limited in size to three by three until Yang Hui’s work in the thirteenth century.

The principal classic source was the Jiuzhang Suanshu (Nine Chapters of the Mathematical Art), composed during the Han dynasty (206 BCE–220 CE). Eminently practical, its nine chapters included 246 problems treating land surveying, proportions of millet and rice, volumes of three-dimensional solids, and tax assessment. Some results were precocious, and the approaches to simultaneous linear equations closely approximated the methods later developed by the German mathematician and astronomer C. F. Gauss (1777–1855). The preeminence of this work sparked during later centuries numerous commentaries elucidating the often obscure explanations. If for some people the commentaries inhibited creative work, other people used the commentaries to introduce original research.

Although geometry largely stagnated at the level of the Jiuzhang Suanshu, Chinese scholars made important advances in algebra and arithmetic. During the fifth century Zu Chongzhi calculated pi to seven decimal places, an achievement first surpassed by the Indian mathematician Madhava around 1400. In 1247 the “0” sign for zero first appeared, and later in the century Guo Shoujing became the first Chinese mathematician to tackle spherical trigonometry. Both developments may have occurred under Arab or Indian influence. Zhu Shijie (1280–1303) presented the French scientist and philosopher Blaise Pascal’s triangle and algebraic methods for solving simultaneous equations and equations of high degree. Only in the eighteenth century would European algebra catch up with Chinese.

## Islamic Mathematics

The Islamic ‘Abbasid caliphate (the office or dominion of a Muslim spiritual leader) (750–1258 CE) sponsored missions to gather astronomical texts (and scholars) from a variety of cultures. Islamic mathematics had two major sets of sources. The first consisted of Persian and Indian sources that featured prominently astronomical tables and an algebraic approach. The second set of sources was Hellenistic, sometimes transmitted through Syrian intermediaries and translations. In their deductive and abstract approach these works in the second, Hellenistic set of sources betray their Greek origins.

The most famous developer of the first Islamic mathematical tradition was al-Khwarizmi (c. 780–c. 850). His treatise Hisab al-jabr w’al-muqabala (Calculation by Completion and Balancing) gives us the word algebra, and his Algorithimi introduced the word algorithm and the Indian positional number system—which had probably first moved out of India with Nestorian (relating to a church separating from Byzantine Christianity after 431) Christians, who found it useful in calculating the date of Easter. In the second mathematical tradition Thabit ibn Qurra (c. 836–901) proved the Pythagorean theorem, attempted a proof of Euclid’s parallel postulate, and made discoveries in mensuration (geometry applied to the computation of lengths, areas, or volumes from given dimensions or angles) and spherical trigonometry.

In addition to preserving these earlier texts in Arabic translation and making further progress within these two traditions, several mathematicians of the Islamic world worked toward a synthesis of Greek deductive geometry and Indian algebra. The fusion of the two streams is best represented by the application of geometry to algebra evident in the geometric solution to quadratic and cubic equations pursued by al- Khwarizmi and the Persian poet and mathematician Omar Khayyam (c. 1040–1123). In addition to pursuing further work on the parallel postulate, Omar Khayyam pursued calendar reform and a theory of proportions useful for the inheritance problems of Islamic law. Jemshid Al-Kashi (1380–1429) relied on a near-circular polygon of 805,306,368 sides to approximate pi to sixteen decimal places. Arab mathematicians enhanced the Indian number system by introducing in the tenth century a small vertical line over the units place to serve the same purpose as the modern decimal point. Nasir al-Din al-Tusi (1201– 1274) anticipated the mathematical improvements of the model of the cosmos developed by the Polish astronomer Nicolaus Copernicus (1473–1543).

## Toward a European Synthesis

Indian numbers reached Europe by the tenth century, were popularized by the Liber abaci (Book of Calculation) of the Italian mathematician Leonardo Fibonacci (1170–1250), and slowly overcame prohibitions by various governments (concerned that the system facilitated fraud) that lasted until King Charles XII of Sweden (1682–1718). The Hindu-Arabic system of our modern numbers attracted merchants with its usefulness in calculations—and attracted swindlers with its easily altered numbers. In Europe resistance to mathematics was not limited to the new number system. The German religious reformer Martin Luther considered mathematics hostile to theology, and the establishment of the Savilian chair in geometry at Oxford University in England prompted concerned parents to withdraw their sons, lest they be exposed to diabolical mathematics.

During the early modern period, as Europeans increasingly exerted influence across the globe, they also began to dominate a mathematical tradition that itself was absorbing compatible elements from other cultures. The dominance of a single stream of mathematics, perhaps fed by many cultural tributaries, is characteristic of modern mathematics.

Europe’s mathematical exchange with China illustrates how global connections created this single stream. In 1607 the Jesuit Matteo Ricci and the Christian convert Xu Guangxi (1562–1633) translated Euclid’s Elements into Chinese. Although Euclid’s deductions contrasted sharply with the inductive approach of the Jiuzhang Suanshu, the translation proved highly influential in subsequent Chinese mathematics. Many of the technical geometric terms coined by Xu remain in use today. Other cross-cultural mathematical connections are less certain. Mathematics from Kerala may have traveled to China, and the Elements may have already been introduced by Arabs. In the other direction, the Jesuits possibly reported some advanced Chinese algebra back to Italy.

Under Chinese imperial sponsorship the text Shu Li Jing Yun (Collected Basic Principles of Mathematics, 1723) marked the end of the integration of Western techniques into Chinese mathematics. During the next emperor’s reign a new closed-door policy restricted access to Western research. Some Chinese worked on critical editions of ancient mathematical texts, while others conducted research in isolation from the West. In 1859 Dai Xu (1805–1860) proved the binomial theorem independently of Newton, whose own 1676 proof had not yet reached China.

Outside the mainstream of Western mathematics stood the Indian prodigy Srinivasa Ramanujan (1887–1920) and his discoveries in the theory of numbers. Although his genius had no contemporary rival, his knowledge of mathematics was limited. Some of his theorems (for example, on the theory of prime numbers) have thus been described as brilliant but wrong. His cavalier attitude toward proof was once attributed to this background, although it in fact fits the Indian pedagogical tradition of leaving demonstration and commentary to students.

The twentieth century has brought a remarkable international tradition of mathematical scholarship often transcending geopolitical divisions. In 1897, 208 mathematicians met in Zurich, Switzerland, as the first International Congress of Mathematics (ICM). Representing sixteen countries, they included twelve from Russia, seven from the United States, and four women. The Cambridge ICM of 1912 counted 82 non-Europeans among its 574 participants. Founded just after World War I, the International Mathematical Union (IMU) excluded the defeated Germany until the IMU adopted a policy of principled neutrality in the 1920s—a policy that would endure despite the challenges of World War II and the Cold War. The mainstream did divide somewhat as Communist and Western mathematicians developed results in parallel, at a lag in either direction, often with different methods. The Commission on Development and Exchange (known before 1978 as the “Commission for Exchange of Mathematicians”) was established in 1959 to facilitate the development of mathematics beyond industrialized countries. The African Mathematical Union formed in 1976, although the exodus of mathematicians from underdeveloped nations to Western research institutes continues to concern some observers.

## Outlook

As historians discover new sources and question old attitudes, the global history of mathematics will continue to expand. The nature of mathematics perhaps makes it more likely to move across cultural boundaries intact, and serious inquiry into the global dissemination of mathematical ideas has only begun. Unfortunately, because so much mathematics has evolved in traditions that do not value the citation of sources, exact lines of transmission will be hard to re-create.

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