Ahmes
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Ahmes (c. 1680 BC-c. 1620 BC) was an Egyptian scribe who lived during the Second Intermediate Period. A surviving work of Ahmes is part of the Rhind Mathematical Papyrus now located in the British Museum (Newman, 1956). Ahmes states that he copied the papyrus from a now-lost Middle Kingdom original, dating around 1650 BC. The work is entitled Directions for Knowing All Dark Things and is a collection of problems in arithmetic, algebra, geometry, weights and measures, business and recreational diversions. The 51 member 2/nth table and the following 84 problems were presented with solutions. But, the scribe only offered brief notes citing the why and how of his often hard to read steps. However, bringing in additional documents like the Akhmim Wooden Tablet, Egyptian Mathematical Leather Roll, Reisner Papyrus and the Moscow Mathematical Papyrus a broader view of Ahmes's math is being found. For example, the 2/nth table may have generally converted 1/p, 1/pq, 2/p, 2/pq and higher vulgar fractions into exact Egyptian fractions series. One, or more, standardized conversion methods were employed. A generalized multiple method, as discussed in the Liber Abaci may have allowed a singular conversion method to be used. If additional conversion methods were employed by Ahmes, the Egyptian Mathematical Leather Roll methods may have been known by Ahmes.
On a broader level, considering the RMP and its parent documents the Egyptian fraction notation was birthed around 2,000 ADE, most likely as a method to replace the Old Kingdom's binary fraction Horus-Eye notation, and its awkward round off system. Finally, considering the connections provided by all the Middle Kingdom texts, the why's and how's of the mathematics that Ahmes drew upon is finally coming into focus. Ahmes' methods, as taught to him, and followed by later scribes always wrote vulgar fractions in exact ways, never rounding off when rational numbers were involved.
Yet, there were times when Ahmes did round off. Ahmes states without proof that a circular field with a diameter of 9 units is equal in area to a square with sides of 8 units (Beckmann, 1971). This method shows that Ahmes tried to 'square a circle', rather than using our modern Greek definition of the area of a circle: pi*r^2. In modern notation Ahmes' method wrote out:
This method leads to a value of pi This irrational number pi approximation reached beyond the rational number domain of Egyptian mathematics, but failed,as anyone would at any time, when compared to modern standards of approximations. This early approximation of pi was consistently used to compute the volume of a hekat, and its many sub-units, including the hin, ro, and dja as recorded in the RMP, Akhmim Wooden Tablet and 2,000 later medical prescriptions reported in the medical texts, information that 2002 work by Tanja Pommerening and others have been reporting after 2001 as new metrological interpretations of the ancient texts, finally being able to translate the information in modern scaled units of measure.
