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Архитектура (936) 
THE ANCIENT ORIENTAL MATHEMATICS
Ø 1) Answer the questions: a)There are four parts of the world: the West, the East, the South, and the North. “The Orient” also means a part of the world. What part of the world is synonymous with “the Orient”? b)What countries of the ancient Orient do you know? Choose from the following: Babylonia, India, Egypt, Germany, Mesopotamia, Russia, Sumeria, Persia.
Ø 2) Read the text and find: a)the date of writing the Papyrus of Rhind and the Moscow Papyrus, b)the date of King Hammurabi’s reign in Babylon, c)the reason for the origin of mathematics, d)the characteristics of mathematics in the Papyrus of Rhind and Moscow Papyrus, e)the difference between Egyptian and Mesopotamian mathematics, f)the reasons for the use of 60 rather than 10 as a unit for time and circle division.
During the fifth, fourth and third millennium B.C. newer and more advanced forms of society evolved from wellestablished Neolithic communities along the banks of great rivers in Africa and Asia, in subtropics or nearly subtropics regions. These rivers were the Niles, the Tigris and the Euphrates, the Indus and later the Ganges, the Huang Ho and later the Yangtse. These territories became centers of civilization. Oriental mathematics originated as a practical science in order to facilitate computation of the calendar, administration of the harvest, organization of the public works, and collection of taxes. The initial emphasis was on practical arithmetic and measurement. Arithmetic evolved into algebra, and measurement developed into the beginnings of a theoretical geometry. The knowledge of Oriental mathematics is very sketchy. The mathematics of Babylonia and Egypt may be considered the most representative of the ancient orient mathematics because there exists a certain consistency in the factual character of the Babylonian and Egyptian texts throughout the centuries. Most of our knowledge of Egyptian mathematics is derived from two mathematical papyri: the Papyrus of Rhind, containing 85 problems and written about 1650 B.C.; and the Moscow Papyrus, perhaps two centuries older, containing 25 problems. The mathematics in these papyri is based on a decimal system of numeration with special signs for each higher decimal unit – a system with which we are familiar through the Roman system which follows the same principle: MDCCCLXXVIII = 1878. The most remarkable aspect of Egyptian arithmetic was its calculus of fractions. All fractions were reduced to sums of socalled unit fractions. The Papyrus Rhind has a table giving the equivalents in unit fractions for all odd “n” from 5 to 101. This work with unit fractions had been practiced for thousands of years, not only during the Greek period, but even during the Middle Ages. It should be noted that all texts point to an Egyptian mathematics of rather primitive standards. Mesopotamian mathematics reached a far higher level than Egyptian mathematics ever obtained. Already the oldest texts, dating from the latest Sumerian period (the third dynasty of Ur, 2100 B.C.) show keen computational ability. These texts contain multiplication tables in which a welldeveloped sexagesimal system of numeration was added to an original decimal system. However, this was not their most characteristic feature. Whereas the Egyptians indicated each higher unit by a new symbol, the Sumerians used the same symbol but indicated its value by its position. Such a system had enormous advantages for computation, as we can see when we try to perform a multiplication in our own system and in a system with Roman numerals. This whole system seems to have developed as a direct result of the technique of administration, as is indicated in thousands of texts dating from the same period dealing with the delivery of cattle, grain, etc., and with arithmetical work based on these transactions. Eventually a special symbol for zero appeared, but much later, in the Persian era. Both the sexagesimal system and the place value system remained in the permanent possession of mankind. Our present division of the hour into 60 minutes and 3600 seconds dates back to the Sumerians, as does our division of the circle into 360 degrees, each degree into 60 minutes and each minute into 60 seconds. There is a reason to believe that this choice of 60 rather than 10 as a unit occurred in an attempt to unify systems of measure, although the fact that 60 have many divisors may also have played a role. As to the place value system, the permanent importance of which has been compared to that of the alphabet (both inventions replaced a complex symbolism by a method easily understood by a large number of people), its history is still considerably obscure. The next group of cuneiform texts dates back to the first Babylonian Dynasty, when King Hammurabi reigned in Babylon (1950 B.C.) and a Semitic population had subdued the original Sumerians. In these texts we find arithmetic evolved into a well established algebra. Although the Egyptians of this period were only able to solve simple linear equations, the Babylonians of Hammurabi’s days were in full possession of the technique of handling quadratic equations. They solved linear and quadratic equations in two variables, and even problems involving cubic and biquadratic equations. The strong arithmeticalalgebraic character of the Babylonian mathematics is also apparent from its geometry. The texts show that the Babylonian geometry of the Semitic period was in possession of formulas for the areas of simple rectilinear figures and for the volumes of simple solids, although the volume of a truncated pyramid had not yet been found. The socalled theorem of Pythagoras was known, not only for special cases, but in full generality. The main characteristic of this geometry was, however, its algebraic character. This is equally true of all later texts, especially those dating back to the third period, that of New Babylonian, Persian, and Seleucid eras (from 600 B.C.  A.D. 300).
