RMP 69 and the Berlin Paprus proportion method

RMP 69, and the Berlin Papyrus Proportion Method

This PM entry discusses ancient scribal methodologies by translating ancient unit fractions into modern rational numbers.

There are three reasons for modern students of Egyptian mathematics to read the 1650 BCE Rhind Mathematical Papyrus(RMP)'s problem 69 in the context of solving two Berlin Papyrus second degree equations.

The RMP problem converted 3 1/2 hekats of grain that made 80 loaves of bread into a pesu unit. Scholars Schack-Schackenburg, early on, commented on a proportion method that a Berlin Papyrus scribe, and Ahmes, the RMP scribe, utilized as a common mathematical tool.

A. First, to understand the shorthand of Ahmes, the RMP scribe, a three phase proof submitted by Ahmes will be analyzed. Initially 3 1/2 hekats of grain meal, that produced 80 loaves of bread, were combined into pesu units for distribution purposes. Ahmes' first phase calculated the pesu as a rational rational number by applying a proportion.

Ahmes conversion of 7/2 hekat, making 80 loaves of bread, to 22 18/21 pesu was achieved by inverting 7/2 to 2/7 and multiplying by 80, reporting:

80 times 2/7 equaled (160/7) pesu and (22 + 2/3 + 1/7 + 1/21) pesu

by applying the Old Kingdom duplation multiplication method.

Modern conversions of hekat, loaf, and pesu date to modern rational numbers discussed the same proportion method used in the Berlin Papyrus. In modern fractions the Berlin scribe found two squared areas, one 10 cubit by 10 cubits, and the second 20 cubits by 20 cubits, by considering the proportions: 1: 3/4, and 2: 1/3.The two Berlin Papyrus problems solved second degree equations

B. Ahmes proved the answer by returning its pesu unit fractions to 80 loaves of bread. This was done by:

(22 + 18/21) pesu times 7/2 equals 80 loaves of bread,

by applying the Old Kingdom duplation method.

C. The second phase of Ahmes' discussion links two Berlin Papyrus solutions to two squares equal to 100 and 400 cubits squared, with x proportional to y by 1: 3/4.

Gillings recorded the proportional relationship as

4x + 3y = 0

3. The BP scribe avoided modern thinking by reducing:

4x = 3y

in the A = 100 problem to:

4. x = (3/4)y

D. for the A = 400 problem considered

1. 2x = (3/2)y

2. The BP scribe proved that either x or 2x proportional solutions were valid by applying the well known proportional method, a direct analogy to the pesu method.

E. The third phase of RMP 69 multiplied 14 ro times 80 to shows that one loaf of bread's ro value was properly stated to 80 loaves in hekat units. Ahmes did that when he wrote:

1. 14 ro as 1/32 hekat + 4 ro

2. 28 ro as 1/16 1/64 hekat + 3 ro

4. 56 ro as 1/8 1/32 1/64 hekat + 1ro

8. 112 ro as 1/4 1/16 1/32 hekat + 2 ro

16. 224 ro as 1/2 1/8 1/16 hekat + 4 ro

32 448 ro as 1 1/4 1/8 1/64 hekat + 3 ro

64. 896 ro as 2 1/2 1/4 1/32 hekat + 1 ro

80. 1220 ro as 3 1/2 hekat

The above reports the obvious data, Scholars have reported this data over the years.

What has not been reported was the meta level of hekat unity (64/64) divisions by n in terms of

(64/64)/n

a method reported in the Akhmim Wooden Tablet that set n = 3, 7, 10, 11 and 13.

What was n at every stage of the RMP 69 duplation proof?

Begin with 14 ro = 14/320 = 7/160

that meant that multiplier 7/160 was understood by Ahmes to be the diviSor 160/7, or

1. n = 160/7

2. n = 320/7

4. n = 640/7

8. n = 1280/7

16. n = 2560/7

32 n = 5120/7

64. n = 10240/7

80. n = 12800/7

F. Ahmes followed the 1825 BCE Akhmim Wooden Papyrus scribe and the Berlin Paprus scribe to solve RMP 69. Middle Kingdom Egyptian scribes scaled the hekat unit to 320 ro units, easing the meta level of ancient expected (theoretical) and actual (practical) calculations, details that are clearly reported in RMP 69, when read in the context of the Berlin Papyrus.