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The Exact Sciences
Mathematics and
Measurement Mathematics served basically utilitarian purposes in Egypt and does not seem to have been regarded as a theoretical science. Only a few mathematical documents survive, including four papyri, a manuscript on a leather roll, and two wooden tablets; the most famous is the Rhind Papyrus. The ancient Greeks considered the Egyptians skilled in mathematics and astronomy, but Egyptian geometry and arithmetic were primarily used for practical applications. The Egyptians are not credited with the abstract thought associated with algebra and equations, which were probably invented by the Babylonians, but they could construct pyramids, obelisks, and rock-cut tombs. This demonstrates that they knew how to draw up plans, obtain accurate measurements, and devise methods for transporting materials and raising and setting in place very large blocks of stone. The arithmetic was clumsy yet effective in being able to yield correct results. The Egyptians used a decimal system of counting. For cardinal numbers they used the powers of ten (corresponding to units, tens, hundreds, and thousands, etc., in the modern system). Vertical strokes indicated the numbers from 1 to 9, while phonetic signs were used for 10, 100, and 1,000; a finger represented 10,000, a tadpole was 100,000, and a kneeling god upholding the sky with his raised arms expressed 1 million. The highest numbers would be written first, followed in order by the lower numbers, ending with the units. Thus, for 1,364 there would be one “thousand” sign, followed by three “hundred” signs, followed by six “ten” signs, and then four units. The numerals were placed after the noun, which usually took the singular form (i.e., year 18 = 18 years). The ordinal numbers (first, second, third, etc.) followed the noun, and if the noun was feminine the ordinal would take a feminine ending (e.g., lord first = the first lord). The Egyptians used addition and subtraction as their basic arithmetical system. They did not have any multiplication tables but used fairly complicated methods of multiplication and division. They either multiplied or divided by ten or, where this was not possible, they had a system whereby they multiplied or divided by two. The Egyptians also represented some fractions and proportional parts, but they usually only wrote fractions in which the numerator was one (e.g., one-eighth), although occasionally the fractions two-thirds, four-fifths, and five-sixths are also found. Although their methods were somewhat tortuous, they could calculate the area of a square and a rectangle (by multiplying its length by its width), and they could measure the surface of a trapezoid. Most significantly, however, they devised a method of calculating the area of a circle according to the length of its diameter, which they obtained by constructing a square, the sides of which were equal to eight-ninths of the diameter. As well as being able to assess the angle of a pyramid they could measure the volume of a pyramid, a truncated pyramid, and a cylinder a skill which helped them to construct pyramids and columns. The Egyptians also faced logistical problems in other areas. They had to evolve systems to measure estates of land so that they could calculate taxes and to organize and distribute food and other supplies to large workforces and to the army. For area measurements they used the setat (100 square cubits), the land mile (1,000 square cubits), and the atour (a square with sides each of one lineal atour). Thus, the Egyptians certainly possessed an organized system of mathematical knowledge, but they used practical experience rather than reasoning skills to solve problems. Although they started with elementary concepts, they soon developed systems that coped with some complicated arithmetical and geometrical problems, and they made use of fractions. .
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Calendar
The Egyptians developed a calendar based on the natural cycle of the agricultural seasons. It was divided into three main periods: the inundation season (akhet), which lasted for one-third of the year; the sowing and growth of the crops (perit); and the harvest (shemu). These three seasons each consisted of 120 days and were divided into 4 months of 30 days; at an early date 5 extra days were added to the calendar to bring the yearly total to 365 days. The inundation never occurred on a fixed date each year, so the Egyptians chose another natural event that would be readily visible to mark the beginning of each year. This was the appearance or rising of the Dog Star, Sirius (Egyptian: Sopdet; Greek: Sothis). The calendar of 365 days was actually too short, a fact that would have become evident after each four successive years, but they did not choose to solve this by adding an extra day every four years. The civil (calendar) year and the agricultural year were closely interwoven; in fact, the start of the civil year coincided with the beginning of the agricultural year. In addition, years were counted from the beginning of each reign; therefore, year 6 of Tuthmosis III refers to the sixth year of his reign. Records and historical documents were kept in terms of each king’s reign. .
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Astronomy
The Egyptians were also noted astronomers who distinguished between the “imperishable stars” (the circumpolar stars) and the “indefatigable stars” (the planets and stars not visible at all hours of the night). They used stellar observations to determine the true north and were able to orientate the pyramids with great accuracy, although the later temples were not given an exact orientation. Each temple was possibly aligned toward a star that had a particular association with the deity resident in that building. Despite these astronomical observations, however, the Egyptians believed that the earth was flat suspended in the midst of a circular ocean; the upper half of this represented the sky while the underworld, the dwelling place of the dead, was situated in the lower part of the circle. .
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The Exact Sciences
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