The clay tablet BM 85200+ containing 36 problems of this type, is the earliest known attempt to set up and solve cubic equations. If problems involving the area of rectangles lead to quadratic equations, then problems involving the volume of rectangular excavation (a "cellar" ) lead to cubic equations. In Berriman gives 13 typical examples of problems leading to quadratic equations taken from Old Babylonian tablets. 5 hours, 25 minutes, 30 seconds, is just to write the sexagesimal fraction, 5 25 60 30 3600 5 \large\frac - \large\frac b 2 x = ( 2 b ) 2 + c − 2 b . This form of counting has survived for 4000 years. The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. For more details of the Babylonian numerals, and also a discussion as to the theories why they used base 60, see our article on Babylonian numerals. It was a positional system with a base of 60 rather than the system with base 10 in widespread use today. The Babylonians had an advanced number system, in some ways more advanced than our present systems. From the mathematical point of view these problems are comparatively simple. They are YBC 4666, 7164, and VAT 7528, all of which are written in Sumerian. There are several Old Babylonian mathematical texts in which various quantities concerning the digging of a canal are asked for. The rulers or high government officials must have ordered Babylonian mathematicians to calculate the number of workers and days necessary for the building of a canal, and to calculate the total expenses of wages of the workers. It was an important task for the rulers of Mesopotamia to dig canals and to maintain them, because canals were not only necessary for irrigation but also useful for the transport of goods and armies. These are discussed in where Muroi writes:. For example we mentioned above the irrigation systems of the early civilisations in Mesopotamia. Many of the tablets concern topics which, although not containing deep mathematics, nevertheless are fascinating. The later Babylonians adopted the same style of cuneiform writing on clay tablets. It was the use of a stylus on a clay medium that led to the use of cuneiform symbols since curved lines could not be drawn. Their symbols were written on wet clay tablets which were baked in the hot sun and many thousands of these tablets have survived to this day. The Sumerians had developed an abstract form of writing based on cuneiform (i.e. However the Babylonian civilisation, whose mathematics is the subject of this article, replaced that of the Sumerians from around 2000 BC The Babylonians were a Semitic people who invaded Mesopotamia defeating the Sumerians and by about 1900 BC establishing their capital at Babylon. The Sumerians, however, revolted against Akkadian rule and by 2100 BC they were back in control. The Akkadians invented the abacus as a tool for counting and they developed somewhat clumsy methods of arithmetic with addition, subtraction, multiplication and division all playing a part. Around 2300 BC the Akkadians invaded the area and for some time the more backward culture of the Akkadians mixed with the more advanced culture of the Sumerians. Writing developed and counting was based on a sexagesimal system, that is to say base 60. This was an advanced civilisation building cities and supporting the people with irrigation systems, a legal system, administration, and even a postal service. The region had been the centre of the Sumerian civilisation which flourished before 3500 BC. Here is a map of the region where the civilisation flourished. The Babylonians lived in Mesopotamia, a fertile plain between the Tigris and Euphrates rivers.
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