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Topic: RMP 53, 54, 55
Replies: 1   Last Post: May 27, 2009 12:55 PM

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 Milo Gardner Posts: 1,105 Registered: 12/3/04
RMP 53, 54, 55
Posted: May 27, 2009 8:14 AM

Dear Forum members:

Translating RMP 53 as three separate problems, two triangle areas and an area of undefined shape, consider

http://planetmath.org/encyclopedia/CubitsEgyptianGeometryAreasCalculatedIn.html by:

A scribal area of triangles and other shapes area calculation method is reported by three Rhind Mathematical Papyrus problems RMP 53-55. The scribal geometry utilized quotients and remainders in an arithmetic context that was looked like scribal weights and measures, algebra, and/or 2/n tables calculations.

RMP 53 calculated the area of two triangles, of 45/8 setat and 63/8 setat, and a third area of an undefined shape by the note, 1/10 of 1 3/8 mh added to 10 cubits of land (COL) is the desired area. A setat was 100 cubit by 100 cubit, or 10,000 square cubits. A cubit of land (COL), or mh, was one cubit wide by 100 cubits long, or 1/100 setat.

The first triangle had an altitude of 5 khet and a base of 9/4 khet. Using the area of a triangle formula, 1/2 the base times the altitude, 5*(9/4)*(1/2)= (45/8) = 5 5/8 setat.

The second triangle had an altitude of 7 khet and a base of 9/4 khet. Using the area of a triangle formula, 1/2 the base times the altitude, Ahmes calculated 7*(9/4)*(1/2) = 63/8 = 7 7/8 setat

The third calculation found the area of undefined shape discussed by:

11/8 mh = 110/8 mh + 10 mh = 23 3/4 mh = 1/8 setat + 11 1/4 mh

since 12 1/2 mh = 1/8 setat.

Scholars have suggested that a truncated pyramid or a triangle defined the third shape.

To assist the decoding of the third RMP 53 area RMP 54, and RMP 55 setat and mh data have been consulted.

RMP 54 partitioned 7/10 setat by 10, 5, 2 1/2 and 1 1/4 segments. Proof was provided by multiplying one setat by 7/10, 14/10, 28/10 and 56/10 within a quotient and remainder context. A quotient setat and a scaled remainder mh were scaled as the 2/n table and a ro unit in hekat (volume unit) were scaled, by writing:

a. (7/10)*(4/4) = 28/40 = (24 + 3)/40 = 3/8 setat + 300/40 mh = 5/8 setat + 7 1/2 mh

b. (14/10)*(4/4) = 56/10 = (55 + 1)/40 = 11/8 setat + 100/4 mh = 1 3/8 setat + 2 1/2 mh

c. (28/10)*(2/2) = 56/20 = (55 + 1)/20 = 11/4 setat + 100/20 mh = 2 3/4 setat + 5 mh

d. (56/10) = (55 + 1)/10 = 11/2 setat + 100/10 COL = 5 1/2 setat + 10 mh

Ahmes may have also made calculations thinking in mh unuts. For example,

Ahmes shorthand partition of 7/10 setat, (1/2 + 1/5) setat, may have focused upon 1/5 setat written as 20 mh. Knowing 12 1/2 mh was 1/8 setat, an answer may have been recorded by:

(1/2 + 1/5)setat = (1/2 + 1/8 + (20 - 12 1/2 mh) = 5/8 setat + 7 1/2 mh.

RMP 55 takes 3/5 of 5 setat to obtain 3 setat by three steps:

a. 1/2 setat + 10 mh

b. 1 1/8 setat + 7 1/2 mh

c. 1 3/8 setat + 2 1/2 setat

d. adding steps a. and c, knowing that 12 1/2 mh = 1/8 setat

(1/2 setat + 10 mb) + (1 3/8 setat + 2 1/2 mh) = 2 7/8 setat + 12 1/2 mh = 3 setat