The Rhind Mathematical dates back to about 1650 BC and is named after the Scottish antiquarian Alexander Henry Rhind who purchased the papyrus in 1858 in Luxor, Egypt, which was apparently found during illegal excavations. It is one of the main sources of our knowledge of Egyptian mathematics. It is also called the Ahmes Papyrus, as Ahmes or Ahmose or A’h-mose was the Egyptian scribe from the Second Intermediate Period and the beginning of the Eighteenth Dynasty who is credited for copying a set of mathematical procedures for this manuscript. It contains a complicated formula which is not perfect, but it does derive prime numbers. The scribe Ahmes states that he copied it from an earlier document dating from the XII-th dynasty from around 1800 BC. It states the following in the beginning, “Correct method of reckoning, for grasping the meaning of things and knowing everything that is, obscurities and all secrets.”

The Rhind Papyrus gives documented proof that the Egyptians were able to do more than just counting numbers. It contains explicit mathematic demonstrations of how multiplication and division were done. This is practical information communicated via example on how to solve specific problems. It also contains evidence of unit fractions, composite and prime numbers, arithmetic, geometric and harmonic means, and how to solve first order linear equations as well as arithmetic and geometric series, and it gives us an ancient estimation for pi, which it is fairly accurate.

The Egyptian measurement of area is termed the ‘setat’ and this was defined by a square with sides being 100 cubits long. Problem 50 of this document questioned how the area of a circular field could be determined if the diameter was known. The Egyptians used a khet for measurement of length and this is about 50 meters. This circular field had a diameter of 9 khet and Ahmes’ stated that the way to solve this is by: Taking away 1/9 of this area, giving you the remainder of 8. Then use solve the area using this 8 as being a side of a square, so you multiply 8 times 8 you will get an area of 64 setat. The solution for this problem of determining the area A of a circle that has a known diameter *d* can be stated with a modern formula, that looks like this: A = (*d* – 1)^{2}. This formula when the diameter is 9 is actually a very close approximation to the modern formula that we use for area of a circle being A = πr^{2}. The problem with this formula is that it only works well for a limited range of diameters, as you can see in the table below. If the diameter is less than 6 or more than 11, you are not getting that good of an estimate.

Circle Diameter |
6 |
7 | 8 | 9 | 10 |
11 |

Ahmes A = (d – 1)^{2} |
25 |
36 | 49 | 64 | 81 |
100 |

Modern A = πr^{2} |
28.26 |
38.47 | 50.24 | 63.6 | 78.5 |
94.99 |

I always learn something I didn’t previously know when I read your posts.

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I guess that is a good thing.

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It…for me, anyway.

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If I am correct the geometry part was primarily for agriculture, but I am still intrigued by how these ancient civilizations did those things. Also khet became so widespread that in many languages it became synonymous to agriculture instead of a unit.

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I am always surprised with your vast extent of knowledge.

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You are knowledgeable Jim, me Nah not even an iota, compare to yours.

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