Arithmetic based om geometry, does that exist?


This article is an analysis of problem number 48 in the Rhind Mathematical Papyrus (RMP). Let's look at the perimeter of an geometric drawing. The perimeter can assume several forms, from a line half the length of the perimeter to several right angled figures and finally the circle which “cowers the largest area of them all. This article will show that the diameter of 9 units, mentioned in problem 48, is possible to relate to the value 28/7 as a side in a square with the perimeter 4 x 7 = 28 units. Shortly: PR 48 seems to be a pedagogical example which is indicating a practical method to determine an approximation of the area of the circle. The table to the right shows one can distribute two perimeters consisting of 16 and 13 units according the formula 4a/2 = 16 or 4a/2 = 13.

There is a rather extensive analysis at this address: http://www.seshat.ch/home/rhind1.htm on the problems in the PR. The document got its name due to the fact that a person named Henry Rhind, 1864 sold it to the The British Museum where the majority of papyrus are kept.

In an article at http://numberwarrior.wordpress.com/2008/03/05/on-the-egyptian-value-for-pi/

the author, Jason Dyer, expresses the opinion that problem 48 relates to a comparison between a squaare and a circle. and asks the question: How did the Egyptians discover the procedure for working with circles in the first place, and a very crucial question: Did the Egyptians really know anything about the constant π? One must remember that RMP is dated to 1650 BC and that the content of RMP is a copy of an older document, according to the writer A´h-mosé, who did the copying. The RMP is divided into 87 problems of which some, the last three are damaged to such an extent that it seems to be impossible to find out what kind of problems the text are treating. There are three problems that seems to relate to the circle; RMP# 41, RMP# 48 and RMP #50. This article will mainly deal with problem #48.
From this simple sketch, from RMP, one may draw the conclusion that the problem have the purpose to compare the inscribed circle and the square. I have some doubt, as I precept that the purpose is to compare a right angle with the sides equal to 9 x 7 = 63, which would be quite a good approximation of the area in a circle with a diameter of 9 units. Some think the picture in RMP 48 is depicting a regular octagon as this geometrical figure is favorable if one want to count squares in a grid of squares 1 x 1.

Here is a copy of an English translation found at the site mentioned above:
The translation goes roughly:

Example of finding the area of a round field with a diameter of 9 khet. What is its area?Take away 1/9 of its diameter, namely 1. The remainder is 8.
Multiply 8 times, making 64. Therefore the area is 64 setjat."

If you find it interesting please download this document:
http://www.ipernity.com/doc/kgesperanto/14122281