in Predynastic Egypt - Page 28
The two circular @-shaped hieroglyphs over the nine decimal hieroglyphs for the decimal 1s (ones) are read incorrectly twice by the Egyptologists. On the one hand, they erroneously see these hieroglyphs as a decimal part of the number below. That is wrong. If these two hieroglyphs were integral parts of that number, the Pharaohs would have written them that way. The Egyptologists argue that the unusual numeric Pharaonic notation found here was thus adopted for "reasons of beauty". Nonsense.
Erroneous is also the reading of the @-shaped hieroglyphs as 100’s, which in fact they were in much later dynasties. Here they stand for 60’s, i.e. 2 x 60 days = 120 intercalary (leap) days. It appears as if this hieroglyph might derive from the similar Sumerian numeric symbol for 60 (or vice versa). We must emphasize that there is no dispute with the fact that the @-shaped hieroglyph represented 100s and that the nail-formed hieroglyph represented 10000s, but in much later dynasties. However, in Khasekhemwy’s day, these numbers were not yet decimally standardized. In 2638 B.C., their decimal value depended on their placement only.[57]
Duncan J. Melville writes that the Sumerian locking of symbols to specific fixed decimal place values occurred only at the end of the 3rd millennium B.C. (2000 B.C.). Prior to that, there was much variation:[58]
"By about 3000 BC, the Sumerians were drawing images of tokens on clay tablets....
Ten cones equaled one small circle, six small circles equaled one big cone, ten big cones was a big cone with a circle inside it, six of those was a large circle and ten large circles was given by a large circle with a small circle inside...."
[Here we have a drawing similar to one in Melville at this juncture:]
"... a single sign might be used in several systems, where it could mean different multiples of the base unit. In particular, the small circle could mean 6, 10 or 18 small cones, depending on context....
The final step in this story, occurring probably some time in the Ur III period, right at the end of the third millennium, was the introduction of a sexagesimal place value system.
The price paid [for this system] was that a vertical wedge could now mean 1, or 60 (6x10), or 3600 (60x60), and so on. Its actual value was determined by its place." [emphasis added]
Surely it was similar in the Old Kingdom of the Pharaohs. 4 – 7 – 9.
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[57] See Duncan J. Melville, Bibliography of Mesopotamian Mathematics and Duncan J. Melville, Sumerian metrological numeration systems.
[58] Duncan J. Melville, Sumerian metrological numeration systems.
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