Rhind Mathematical Papyrus
The Rhind papyrus is an ancient Egyptian, written on papyrus treatise on various mathematical topics, which we refer to as arithmetic, algebra, geometry, trigonometry, and fractions today. He is next to the slightly older, but less extensive papyrus Moscow 4676 as one of the main sources for our knowledge of mathematics in Ancient Egypt and is dated to about 1550 BC.
Discovery
The papyrus is named after the Scottish lawyer and antiquarian Alexander Henry Rhind, who purchased it in 1858 in Luxor, Upper Egypt. The documents were probably little previously found in or near the Ramesseum in illegal excavations on the opposite Luxor west of the Nile lying area of Thebes, precise circumstances are not known.
Details
The Rhind papyrus was made in the 16th century BC during the Second Intermediate - introduction is the 33rd year of the reign Apopi, a king of the 15th dynasty of the Hyksos, specified as a date - and is in substantial part as the copy of a two centuries old papyrus viewed, which is probably from the reign of Amenemhat III. the 12th Dynasty in the Middle Kingdom came. The copyist, a scribe named Ahmose (after an earlier transcription also Ahmes ), used the hieratic writing and picked up some values and procedures listed in red instead of black ink out, for example, sets of dividers. Today, the Rhind papyrus in the form of fragments of about 5 meters long and about 32 cm wide scroll is present, which is written on both sides, and plays various mathematical problems with exemplary solutions, in total, depending on how you count 84 or 87 tasks. The text was only at the end of the 19th century AD are deciphered and translated, his mathematical statements are decrypted and opened up since the beginning of the 20th century.
Content, the manuscript can be divided into three departments. A longer table n 3-101 for all odd numbers the fraction 2 / n is as a sum of unit fractions, found in the first part, which deals with 40 arithmetic and algebraic problems. The second part presents 20 geometrical problems and treated room contents and surface areas of different characters and the ratio of height to side of the body of a pyramid as their inclination. Two dozen more problems form the third part, based on calculations in addition to the production of bread and beer, as well as to the feeding of poultry and cattle, among others, a mystery object is reproduced here to cats and mice.
Approximate calculation of the area of a circle
In the 48th task Ahmes describes how he calculates the area of a circle is inscribed in a square with a side length of 9 units. From today's perspective, this can be interpreted as giving an approximation of the circle number.
These three Ahmes divides first the sides of the square and thus wins nine equal smaller squares with the side length of 3 units. Then he cuts from the four corner cells away each half and comes over to the figure of an irregular octagon. This octagon is made up of five full and four half to the total of 7 of the small squares, each with 32 = 9 area units together and so has the surface area of 7 • 9 = 63 square units. It is obviously only slightly smaller than the circuit - for the area of which increases Ahmes therefore the content of 64 = 8 • 8 square units, which is not small.
Thus, the area of a circle with the diameter 9 is set equal to the area of a square with sides of length 8. It follows approximately for the content of the circular surface with a radius of 9/2
The value thus determined missed the number by about 0.01890 absolutely and relatively to less than one percent. In the ancient Egyptian number system, this value is not shown in decimal, but as a sum of unit fractions:
The reproduced in the Rhind papyrus method, the wave number can therefore be calculated from the ratio of the areas of the inscribed circle and its circumscribing square,
The a square with 81 units of area inscribed circle actually embraces about 63.617 units of area. In approximation of a circle is here by the set prescribed by Ahmes method based on a square of 9 • 9, mediated by an octagonal figure and its area equal to a square of 8 • 8 - which can be seen as early attempt at squaring the circle well.
Place to keep
The Rhind papyrus or Rhind Mathematical Papyrus (RMP ) is located since 1865 in the possession of the British Museum in London under the inventory numbers PBM 10057 and PBM 10058, apart from some smaller fragments, which were then acquired by Rhind and is now at the Brooklyn Museum in New York are kept.