Struik: A Concise History of Mathematics: The Orient after the Decline of the Greek Society

by Carson Reynolds

Despite Hellenistic influence, Near Eastern thought remained intact, as is evidenced by work in Alexandria, India, and Constantinople. The Byzantine Empire served as a guardian for Greek culture while the Indus region and Mesopotamia became independent. The sudden growth of Islam ended Greek domination. Arabic administration and language competed with and conquered Greek culture in much of the Mediterranean.

As the roman empire declined the center of math research shifted from Alexandria to India and Mesopotamia. The Surya Siddhanta shows an influence of Greek and Babylonian astronomy. Aryabhata (c. 500) and Brahmagupta (c 625) were the best known. Mahavira considered rational triangles and quadrilaterals. General solutions for indeterminate equations of the first degree (ax+by =c) is found in Brahmagupta. Bhaskara admitted negative roots of equations and his Lilavati became a standard text for arithmetic and mensuration. Nilakantha (c. 1500) had already found the Gregory Leibniz series for pi/4.

Our present decimal-position system first appeared in China and was used increasingly in India (c. 595). The word sunya although the use of a dot predates this in Babylonian texts. “0” probably comes from the Greek ouden (nothing). However, in Hindu math, zero was equivalent to 1…9, not just a holder as with the Babylonian dot. Translation of the Siddhantas into Arabic introduced the Hindu system to the Islamic world, where permutations of it (the gobar system made their way to Spain and to the West.

Persia and Baghdad were taken by Arabs, causing Arabic to be instated as the official language, although other cultures remained. Islamic math was influenced by the same factors as Alexandria and India. The caliphs promoted astronomy and math, creating libraries and observatories. Muhammand ibn Musa al-Khwarizmi (c. 825) wrote a book whose Latin translation (Algorithmi de numero Indorum) spread the decimal position. The word “algorithms” is a latinization of his name. Similarly his Hisab al-jabr wal-muqabala (science of reduction and confrontation or science of equations) introduced al-habr or algebra into the lexicon. Although lacking formalism and mostly geometric, his examples (i.e. x^2+10x=39) were a thread appearing in algebras for several centuries. He also included trigonometric tables. His geometry, while simple can be traced to a Jewish text of 150 CE. His work lacked the axiomatic foundation, but was important for the introduction of decimal position to the West.

Arabic scholars also faithfully translated Greek classics into Arabic: Apollonius, Archimedes, Euclid, Ptolemy (Almagest being the familiar name for his Great Collection). Arabic math was particularly interested in trigonometry (sinus is a latinization of the sanksirt jya). Sines were half a chord, and were thought of as lines. Al-Battani provided extensive cotangent tables (for every degree) Abu-l-Wafa inroduced secant and cosecant, and derived the sign theorem of spherical trig. Al-Karkhi (d. c. 1029) was monomaniacal interested in Greek and wrote an algebra inspired by Diophantus and was interested in surds (sq roots).

Omar Khayyam (c. 1038-1123), who lived in northern Persia near Merv, was notable for a reformed Persian calendar with an error of one day in 5000 years (compared to 330 years w/ the Gregorian). His Algebra examined cubic equations and determined root as the intersection of two conic sections. He also introduced a non-Euclidean geometry. Nasir al-din separated trig from astronomy and attempted to prove Euclid’s parallel axiom, which was made use of in Renaissance Europe by John Wallis. Nasir followed Khayyam’s approach to theory of ratio and the irrational. Jemshid Al-Kashi (d. c. 1436) was influenced by Chinese mathematics and knew of (what is now called) Horner’s Method, iterative methods. He also provided the binomial formula for a general positive integer exponent. Al-Kashi had pi to 16 decimals. Ibn Al-Haitham, whose Opticswas influential, solved the problem of Alhazen. He also employed the exhaustion method. Abu Kamil (a follower of Al-Khwarizmi) had influenced on Al-Karkhi and Leonardo of Pisa. Al-Zarqali, was notable for the Toledan tables, which influenced the Alfosine tables, which were authoritative trig tables for centuries.

Chinese mathematics was not isolated. It developed at least by the Han Dynasty, the decimal position system was probably invented there. Pi was found to many decimal places (Liu Hui had two digits, Tsu Ch’ung-Chih, had seven [22/7]). During the Tang dynasty, imperial examinations made use of math books, spurring the printing of Nine Chapters as early as 1084. The Sung dynasty saw greater progress. Wang Hsio Tung exceeded the Nine Chapters by looking at cubic equations of a higher complexity. Ch’in Chui -Shao used successive approximation to solve higher degree polynomials (similar to Horner’s work of a much later date). Yang Hui (c. 1261) used a decimal notation similar to our modern style. He also provided the earliest extant pascal’s triangle. Chi-Shih-Cieh, the most important Sung mathematician, extended “matrix” methods to solve linear equations with several unknowns and of a high degree. The post-Sung period saw a decline, but diffusion of these developments westward.