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Rhind Mathematical Papyrus (RMP): Lesson 50 An excerpt out of the book “The royal Circles” by J.Borbola

Picture 10 . G.Robins a ch. Shute, The Rhind Mathematical Papyrus.

Plate 16 problem 50

Picture 11 August Eisenlohr: Ein mathematisches Handbuch der alten Aegypter. Taf. XVII.

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A. Processing the text

Fist line

First line

Fejen számít felszínt. A csíkon kapni, 9 kötél…

Head-calculating surface. Getting (on) surface (of) 9 kötél <kœtel> (rope)…

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Second line

Line 2/A

Pajta térnek nevezett rakottságok íve, mennyi a felszíne?

Is the diameter of the granary(-yard), how big is its surface?

Line 2B

Körbe osztasz kerek 1/9 ívet, ami 1.

You divide the circle by 9, which is 1

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Third line

3/A line

Vegyétek, ami 8. Számolok

Take it (from 9), which is 8, I’m counting

3/B line

Szükséges a fej, ami 8 szaporítasz 8-cal…

Head needed, which is 8, multiply by 8..

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Fourth line

4/A line

…kapol körívet, ami 64 rakottságok…

You receive a circle-surface, which is 64…

4/B line

Ívében, ami felszín 64

In its extension, it is a surface of 64

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Horizontally organized:

Számit(s), mérj, kapol, kötélben 9

Calculate! Measure! receiving, 9 in kötél.

B. The process of calculation

1 x 8 Körbe/kőláb oszt, király menetű ív, vegyétek, ez 8.

divides circle, surface by royal-course, take it from, it is 8 2 x 16

4 x 32 8 x 64

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C: The end-result

Rakottságok íve, ami marhát csíkoz

6 + 4

64

The depository’s territory, its surface is 64

D. The Hungarian / English reading Fejen számít felszínt, amit a csíkon kapni. 9 kötél a pajta-tárnak nevezett rakottságok íve, mennyi a felszíne? Kőláb oszt kerek 1/9-ed ívet, ami 1. Vegyétek, ami 8.

Számolok, gondolkozz! Ami 8, szaporítasz 8-cal, kapol körívet, ami 64 a rakottságok ívében. Ami felszín, az 64.

Head-calculating surface. Getting (on) surface (of) 9 kötél <kœtel> (rope)… Is the diameter of the granary(-yard), how big is its surface? You divide the circle by 9, which is 1. Take it (from 9), which is 8, I’m counting, Head needed,

which is 8, multiply by 8, You receive a circle-surface, which is 64 in its extension, which is a surface of 64.

Calculate! Measure! receiving, 9 in kötél divides circle, surface by royal-course, take it from, it is 8

The depository’s territory, the surface of it is 64

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E. Notes

The presence of a measurement-unit in this lesson is a remarkable difference

from other lessons. Egyptologists named this in the text well readable sign ‘khet’, which suppose to stand for a unit of length. According to Hannig1 : 1 ḫt = 100 mḥ

(Ellen) = 52.5 m. 1 Elle (cubit) = 52,5 cm, it is called the “royal cubit”. However, Eisenlohr calculated already 130 years ago that this would lead us to

enormous large sizes: “As an example, the 64 khet2 in the lesson 50 are then 64 x

27,56 m2 = 1762,84 are or 17,64 hektares, almost 70 Prussian morgen. It is definitely too large for one piece of a plough-land.”2 (By the way, Eisenlohr’s calculation wasn’t exact. The product of 64 x 27,56 m2 appears in m2, not in ares. Thus it’s change to

are slipped by two decimals. Further more, 27,56 m2 are only the quadrate of 5,25 m, it is just one tenth of 52,56 meters. Interestingly, his end-result of 17,64 hektars, calculated with a khet of 52,56 meters is correct again.)

Let us see, what kind of measure and how large is khet in reality?

We read the signs (16-17) as kötél <kœtel> (rope) instead of khet. As a unit of

length, it could be 5,25 m by measuring 10 cubits of 52,25 cm. (We do not have any proof until now, that the measurement-units followed the decimal system.) The size of

100 cubits - seeing Hannig’s figures above – is not likely, it is uneasy to handle. It is not easy to carry around a rope of 52,25 meters, to span or roll it up particularly on a wet terrain. The borders of the fields were washed away during the flooding season

and had to be repeatedly measured. We can see land-surveyors with rope in their hands on several pictures, two or more persons. (See picture of table XI in Appendix).

The unit 9 khet is well legible in lesson 50. If our guess from above is true, we

receive the result in khet2, that is 64 x (5,25)2 m2 = 1764 m2 This is a plough-land of around 30 x 60 meters (32,8 x 65,6 yard = 2151,7 square-yard). We note here that there may have been other length-measures between the khet/kötél and cubit, like a real kar (arm) or ől (cord or fathom).

The sign of khet appears in the lessons 49-52 with different measures what

renders our investigation more difficult. It characterizes the sides of a rectangle in the lesson 49 combined with the numbers 2 and 10. in In the lesson 51 the base and one hypotenuse of a three-angle are given in khet. At the base the number four

accompanies the khet, while the ten-valued khet-sign of the hypotenuse carries 3

dots at the bottom. In the lesson 52, we can read khet-3, khet-6, khet-4 embedded in the text.3 Therefore, we can’t stop investigating the length of the ‘kötél’ yet.

This lesson introduced the simplest method of calculating the circles surface, thus there is no doubt about the process. Our scribe added explanatory text to the numbers. The result of the surface is with no doubt 64 kötél2.

The column, just before the mathematical work-out, differs somewhat from the previous ones: calculate!, measure!, you get, and 9 kötél are written one beneath the

other. It seems, our scribe shortened the text. Not in the current text, but he writes

here separately that the stone-foot divides and this is the “royal course”.

1 Hannig, op. cit., 1285.

2 Eisenlohr, op. cit., 124.

3 You can check these hieratic signs in Table VI of the Appendix.

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Interestingly, he wrote the number 64 in two different ways in the fourth line. We see first the hieratic signs of 6 and 4, but at the end of the line stands after the 6 the sign 4. The final result is written with 6 and 4 as well.

This supposed to be a simple problem according our geometric knowledge of today.

Summary Everybody can follow the simple method to figure out the circle-surface in this

lesson, thus Egyptologists like to refer to this lesson, when dealing with circle-

calculation. The analysis of the lesson lead to numerous mathematical and geometrical “inventions” and countless variants emerged. Among many of them, this supposed to be the origin of the thesis by Thales and Pythagoras. All these belong

naturally to the attributes of every circle and are valid for the circle of the lesson 50 as well. However, one can’t originate such a conclusion from this lesson.

Its language style and mathematical expressions are identical with those of other

lessons. Its place in the papyrus is not logical. The from the previous ones different measurement-unit, the khet / kötél supposed to be the reason for this.4

With this we finished the lessons of the RMP dealing with circles. As we know, there are no more lessons of this kind in the papyrus.

Egyptologists or anybody else interested in reading ancient Egyptian texts in Hungarian, please contact J.F.T. Borbola, janoslo@kabelfoon.nl. the author of the book “Royal Circles”, (2001).Reading RMP (Rhind Mathematical Papyrus) in Hungarian. Let’s read it together in Hungarian (MMP: Moscow Mathematical Papyrus), (2000). Csillagszoba (Star Chamber), Hungarian reading of the pyramid scripts in Sakkara,

(2004) Az epyitomi ősmagyar nyelv. (The Ancient Hungarian Language of Egypt), (2012). (The solution (Hungarian reading) of Paper 1115 in St. Petersburg. About the assumed story of shipwrecked people) Egyiptomi ősmagyar geometria (Ancient Hungarian Geometry of Egypt) (2013)

(Geometric lessons of RMP dealing with the circle and Pyramid.)

4 We still say in Hungarian „kötelez” somebody to accept an obligation, or the result of an oficial land-

measuring.