Berlin Papyrus
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The Berlin Papyrus 6619, commonly known as the Berlin Papyrus[1] is an ancient Egyptian papyrus document from the Middle Kingdom.[2] This papyrus was found at the ancient burial ground of Saqqara in the early 19th century CE.
The papyrus is one of the primary sources of ancient Egyptian mathematical and medical knowledge[3], including the first known documentation concerning pregnancy test procedures, and is thus part of the medical papyri.
The Berlin Papyrus contains a problem stated as "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other."[4] The interest in the question may suggest some knowledge of the Pythagorean theorem, though more likely the data shows a straight forward solution of two second degree variables stated as one unknown, and not two unknowns, as reported by Scott Williams:
"100 square cubits is equal to that of two smaller squares, the side of one square is 1/2 + 1/4 of the other. What are the sides of the two unknown squares.
In modern terms we would express this as x2 + y2 = 100 and x = (3/4)y. What are x and y? A modern solution in this form might be ((3/4)y)2 + y2 = 100 implies (1 + 9/16)y2 = (25/16)y2 = 100 implies y2 =(16/25)100 = 64 implies y=8 and x= (3/4)8 = 6.
However, most translators believe the egyptians viewed this problem the way we do the simultaneous equations
x2 + y2 = 100 4x - 3y = 0 What are x and y?
Here was their solution. Assume the square of the first side (y) to be 1 cubit. Then the other side (x) will be 1/2 + 1/4. Then y2 = 1, and using Egyptian multiplication we determine x2 with 1 1/2 + 1/4 1/2* 1/4 + 1/8 1/4* 1/8 + 1/16 1/2 + 1/4 1/4 + 1/8 + 1/8 + 1/16 = 1/2 + 1/16
So x2 = 1/2 + 1/16. Thus, x2 + y2 = 1 + 1/2 + 1/16. Now (1 + 1/2 + 1/16)1/2 = 1 + 1/4 and (100)1/2 = 10 (we will discuss square roots later). Divide 10 by 1 + 1/4 and you get 8 (see the method of problem 24). So we get y=8. The Berlin Papyrus contains damage here so we can at best assume the solution for x was to divide 8 by 1/2 + 1/4 (as in the method of problem 24) to achieve x=6.
Berlin Papyrus Problem 2. You are told the area of a square of 400 square cubits is equal to that of two smaller squares, the side of one square is 1/2 + 1/4 of the other. What are the sides of the two unknown squares.
This is analogous to problem 1, ..."
[edit] See also
[edit] References
- ^ Lumpkin, Beatrice, The Mathematical Legacy of Ancient Egypt - A Response to Robert Palter, 2004. National Science Foundation. p17
- ^ Corinna Rossi, Architecture and Mathematics in Ancient Egypt, Cambridge University Press 2004, p.217
- ^ Dr. Williams SUNY-Buffalo
- ^ Richard J. Gillings, Mathematics in the Time of the Pharaohs, Dover, New York, 1982, 161.
[edit] External links
- Simultaneous equation examples from the Berlin papyrus
- Two algebra problems compared to RMP algebra
- Two suggested solutions
- Medicine of the Pharaohs
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