Struik: A Concise History of Mathematics: The Ancient Orient
by Carson Reynolds
5-3 millennium BCE saw the development of neolithic culture in the fertile crescent, the Indus, Ganges, and Huang Ho and Yang-tse Rivers. The structures required to maintain agriculture led to greater administration and with it computation and mensuration. Villages oscillating between anarchy and tyranny were the source of slow, and erratic cultural progress. The static, despotic, and religious nature of early administrations meant that in many middle-eastern and Asian cultures, priests were the carriers of scientific knowledge.
Oriental (this is Struik’s ethnocentric term) mathematics started to deal with practical mattes like computing calendars, administering harvest, and taxation. Arithmetic evolved into algebra (the consequence of better computation and scribes). Mensuration developed into the beginnings of geometry. Chinese and Egyptians were isolated while Mesopotamian and Indian traded. It is easy to differentiate the development of these different cultures, b/c of different symbols. Many cultures claim antiquity, but Mesopotamian has the largest amount of sophisticated extant material (in the form of clay tablets).
The Papyrus Rhind and The Moscow Papyrus give us most of our knowledge about Egyptian math. It was more roman like, a decimal system with special signs. It tended toward addition instead of multiplaction 13×11 was obtained by first multiplying simpler terms and then adding. 13*11 = 1+2+(2*(2*2)) * 11. The fractional system was more remarkable. Everything was represented as unit fractions. 2/n tables were provided to aid decomposition. Algebra was performed witht the aid of the sybol aha or hah meaning heap. Egyptian algebra is known as aha-calculus. Egyptians knew the area of the triangle and had a formula for the diameter of the circle with pi = 3.1605. More remarkable was a value for the frustum of a pyramid. However, despite all of this we should reject claims that Egyptian math was far advanced; it was rather primitive, as was their astronomy.
Mesopotanium math was much more advanced than egyptian math. They had a sexagesimal system which used position for value (as our system does). A blank was used for zero, until a special symbol was developed during the Persian era. Zero was a consequence of position-based systems. Division of the hour and circle into multiples of 6 derive from Sumerians.
Cuneiform tablets from King Hammurabi (c. 1950 bce) show sophistication: solutions to linear equations, quadratics in two variables, cubic and biquadratic equations. The character was strongly arithmetical-algebraic. The Theorem of Pythagoras was known. The square root of two and was approximated to two decimal places and pi two one. (25/8). They also had formulas for compound interest. Ideograms were developed for math concepts. Development was stimulated by contact with other civilizations. Later developments were based not just on Egyptian but Babylonian math.
Proofs and demonstration didn’t occur in ancient oriental math. Instead we’d have recipes. Of course algebra is taught in modern times by memorization of rules.
Hindu math had special signs for each of the numbers 1:1:9;10:10:100;1000:100:?. Sulvasutras contained math rules for religious rights. Some knowledge of the Pythagorean theorem and approximations of fractions existed. Good approximations for the sqrt of 2 exited and pi was given as the sqrt of 10.
Chinese math can be traced through Nine Chapters on the Mathematical Artas wel as the lo-shu (magic square) in the I-ching. Chinese numeration was decimal, numbers were expressed by arrangements of bamboo sticks. The calendar was sexgesimal (60 was Tennyson’s “cycle of cathay). Square and Cubic roots were found, pi was taken as 3. Systems of linear equations were solved using matrix transformations. The tradition of Chinese math wasunbroken and quite stable (or stagnant, depending on one’s point of view). Memorization was the pedagogical principle (some Indian texts were written in rhyme).