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Author Topic:   The Rhind and Moscow papyrus

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Registered: Feb 2003

posted 22 January 2005 02:02 PM     Click Here to See the Profile for ausar     Edit/Delete Message   Reply w/Quote
The Moscow and Rhind Papyrii

[The Rhind Payrus] The Rhind, or Ahmes papyrus is one of our best records of Egyptian mathematics. Its two names come from the fact that it was written by a scribe identified as Ahmes. It was later purchased by a Scottish Egyptologist, Henry Rhind in 1858. It was originally written around 1650 BCE, and is believed to be a copy of an earlier work dating to about 1850 BCE. It currently resides in the British Museum.

[The Moscow Papyrus] Unlike the Rhind papyrus, the author of the Moscow papyrus is unknown. It is slightly older than the Rhind papyrus, and dates to about 1850 BCE. Sometimes referred to as the Golenischev papyrus, after the Russian collector who purchased it, the Moscow papyrus resides in the Moscow Museum of Fine Arts.

Both the Rhind and Moscow papyrii are very long manuscripts, and only portions are shown above. Both were written on sheets of papyrus using cursive hieratic script. Both are compilations of mathematical problems and solutions. The Rhind papyrus contains 85 ancient problems and the Moscow papyrus contains 25 ancient problems. The solutions presented on both are in the form of instructions, and do not include proofs.

Ancient Civilizations, Part 2

Today's presentation began where the last one left off, Egyptian mathematics. Br. Barsky presented us with several papyri containing mathematical computations. One of them, the Moscow Papyrus, contained somewhat of a solution to finding the volume of a truncated pyramid. It was called Problem 14.

Problem 14 contained a step by step set of instructions that eventually ended up with a number that was the volume of the truncated pyramid in question. Upon further examination of the steps and by using variables instead of numbers, the modern day formula for the volume of such a figure was formed. Thus it seems to imply that the Egyptian mathematicians knew how to formulate this rather involved equation.

After a brief commentary about Sunday, 8/25/96, Thomas Muir's birthday, Dr. Barsky's presentation began to generate an overall theme of Egyptian mathematics specifically in reference to the volume of a pyramid. In the scope of this lecture, Dr. Barsky covered the topics of Moscow Papyrus, Rhind Papyrus, the Rosetta Stone, the volume of a pyramid, the area of a triangle, and the area of a circle.

Within this array of topics, I became most curious as to how an ancient civilization could figure out the connection between the area of a square and the area of a circle. I can see from problem forty eight that the calculation or the area of a circle differs by one less than the square in the multiplicand, and if I draw a square around a circle, I can clearly see that the edges of the square protrude from the edge of the circle, but how do they know how much to round the edges? They could see that the length of one side of the square equals the diameter of the circle, which equals nine, and in calculating the area of the square, the diameter equals the multiplicand and the original multiplier. They then would subtract one divided by the diameter from the diameter to calculate the area of the circle,(D-1/D)^2. Why did they choose to do this? I think they stumbled upon it by accident. Maybe one of the priests became intrigued by the fact that by doubling in one direction/multiplying, the numbers increased in the same magnitude that the numbers decreased when doubling in the opposite direction/dividing, ex.(1/8,1/4,1/2,1/1,2/1,4/1,8/1). Since the Egyptians, unlike the Greeks, concerned themselves mostly with formulas, this priest decided he would subtract the reciprocal of the diameter from the diameter to round the edges, D-1/D. He then needed to square the result to polish off the rounding. After using it and noticing that it relatively worked,(even though the result differed from pi multiplied by the radius squared by .415 in problem forty eight), he made a note of it so he could teach others.

This lecture was centered around Egyptian mathematics. They are unique in that we actually have access to some of the original mathematics artifacts. The Moscow Papyrus, Rhind Papyrus, and Rosetta Stone, found by Jean Francois Champollion, and written in three languages, were discussed.

The major focus was on the Egyptian formula for the volume of a truncated pyramid. We discussed possible methods that could explain how they came up with that formula. One interesting point that was made was the fact that we are unsure of what kind of assumptions they used or what knowledge they were actually aware of.

Although the Egyptians came up with the correct formula, the Babylonians were close, but incorrect. Katz brings up the point that although the Babylonians were wrong, the answers they would get would not be that far from the correct one. "It is difficult to see how anyone would realize that the answers were wrong in any case, because there was no accurate method for measuring the volume empirically."(p. 21) On the same page, Katz also notes that the problem with these formulas were related to the amount of people that it would take to build the structures. This remind us that these formulas were developed because of a need to compute certain values quickly, there was a practical use for them. This, I believe, is how many of our equations and formulas came to be. These people didn't just sit around looking for a reason to do math, they were more worried about making it easier to perform the all important tasks of daily living.

Dr. Barsky started out by explaining that the reason he chose to focus on Egypt in our discussion on ancient civilizations was that many of the documents related to the Egyptians are actually originals, or some close facsimile thereof. With many others this isn't the case. For example, the writings of Euclid were copied or translated many times before currently existing documents came to be.

Dr. Barsky explained that the Egyptians did a lot of writing on long scrolls, or papyri. These papyri contained diagrams and text, including writings on astronomy, which was one of the reasons they had for studying math. They wanted to predict what was going to happen. Specific examples mentioned were the Moscow papyrus, dated around 1850 B.C., and the Rhind papyrus, believed to be from around 1650 B.C. Also touched upon was the Rosetta stone, which as a polished stone with writing on it. The writing appears to be the same story written in three different languages or dialects. The Rosetta stone was instrumental in learning to decipher hieroglyphics.

The discussion then turned to finding the volume of a truncated pyramid, which was illustrated in problem 14 of the Moscow papyrus. Dr. Barsky showed how the Egyptians' solution could be reached algebraically, but he seemed to be inclined to believe that they found it as a sum. He demonstrated with Play-Doh that a truncated pyramid could be broken down to a rectangular solid, two triangular solids which, when put together make another rectangular solid, and a smaller pyramid.

How the Egyptians arrived at their formula is open to speculation. But what interests me more than how is why. Katz talks about different ways that ancient civilizations worked toward estimating the area of a circle, including cutting it into wedges and arranging the wedges to approximate a parallelogram, or inscribing and circumscribing polygons, noting that the latter becomes more accurate as the number of sides on the polygon increases.

It's easy to see why area would have piqued the interest of people back then. Katz emphasized this by mentioning that people once believed that the area of a rectangular plot was dependent only on the perimeter, and some people who knew better took advantage of this by selling smaller plots of land for higher prices by measuring only the perimeter. It's also easy to see why astronomy was important. It could help in the creation of calendars, which would be used to find the optimal time to plant crops or breed animals for the production of food. But finding the volume of a truncated pyramid would seem to indicate a desire to obtain knowledge simply for the sake of the knowledge.

This lecture focused primarily on the mathematical knowledge of the ancient Egyptians and the sources from which we gather our knowledge of the mathematics of ancient Egypt. The two main sources of Egyptian mathematical knowledge are known as the Moscow Papyrus and the Rhind Papyrus, from -1850 and -1650 respectively. A problem from the Moscow Papyrus, how to find the volume of a truncated pyramid, was then discussed in detail. Several possible methods of obtaining the volume in question were discussed along with certain problems that might have cropped up in the derivations of the formula.

In order to find volume of the truncated pyramid, it seems obvious that the Egyptians would first have to derive a method of finding the volume of a complete pyramid. An interesting method was discussed in which a cube was divided into three identical pyramidal shapes. Since the Egyptians surely knew the volume of a cube, obtaining the volume of 1/3 of a cube would be simple. One method discussed in detail was finding the volume of a complete pyramid and then subtracting the volume of the "top" piece from the volume of the whole. While this method produced the same "formula" described in the Moscow Papyrus, it involved algebraic techniques that were not likely known at the time.

Mathematics in the Time of the Pharaohs

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posted 22 January 2005 02:16 PM     Click Here to See the Profile for King_Scorpion     Edit/Delete Message   Reply w/Quote
This is one of the reasons why Egypt is talked about so much...and also why conspiracy theorists believe aliens were there...(rolleyes). Egyptians were the top dogs for thousands of years. Rome was really the first Empire to overshadow Egypt. But by the time of Rome, Egypt's time had already come and gone. Even the Romans saw Kemet as an ancient land.

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posted 22 January 2005 05:18 PM     Click Here to See the Profile for King_Scorpion     Edit/Delete Message   Reply w/Quote
C'mon guys let's get some replies. Let's prove the racists wrong. We CAN talk about other things than skin color.

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posted 22 January 2005 09:53 PM     Click Here to See the Profile for supercar     Edit/Delete Message   Reply w/Quote
We've actually touched on this, in an old thread called "Kemetian Mathematics":

However, new insights are always welcome.

[This message has been edited by supercar (edited 22 January 2005).]

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posted 23 January 2005 12:00 AM     Click Here to See the Profile for HERU     Edit/Delete Message   Reply w/Quote
Speaking of the Moscow papyrus, is it true Archimedes' "The Method" has formulas (or what have you) "comparable at every point"?

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posted 24 October 2005 09:00 AM     Click Here to See the Profile for tdogg     Edit/Delete Message   Reply w/Quote

Any new information on these papyrus?

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