# The Rhind Mathematical Papyrus, 1575 BCE

The most famous mathematical work form dynastic Egypt is the Rhind mathematical papyrus, copied by the scribe Ahmes or Ahmose from a papyrus of 1849-1801 B.C.E. It was bought by A. H. Rhind at Luxor in 1858 and is now in the British Museum. In its geometry we can see some understanding of the properties of right-angled triangles, including the simplest ones, which have sides that are integral numbers of units in length (3, 4, 5; 5, 12, 13). Also included is an approximation for π, the ratio of the circumference of a circle to its diameter; the given is (16/9)2 = 3.1604938, too large by about 0.019.

Geometry was equally required for the construction of pyramids. In the first place, care was often taken to achieve orientation north, south, east and west. North—south orientation could easily be obtained by finding the direction of the noonday sun. A vertical pole was set up in the sand as a gnomon. Then the path traversed by the tip of its shadow could be observed. The points A and B, where this intersected a suitable circle drawn around the gnomon, would be joined. Then the line AB was bisected to establish the direction of the sun at mid-day.

Egyptian pyramids were square in ground plan and fully pyramidal in shape. The specification for the gradient, known as the “batter,” of such pyramids was denoted by the hieroglyphic word skd, meaning ratio. It was expressed in terms of the number of palms, in half the length of a side, per cubit of vertical height. The Rhind papyrus is particularly concerned with the geometry of pyramids, measurements being reckoned in royal cubits. An example of finding the batter is- “A pyramid whose vertical height is 93 1/3 cubits. Let me know its batter, 140 cubits being the length of its side.” Half the length of a side is 7 x 70 palms; the batter is therefore

7 x 70/93 1/3 =7 x 210/280=5 ¼ palms;

since 1 palm= 4 finger’s-breadths, this is expressed as 5 palms, 1 finger’s-breath.

The British Museum

Dilke, O.A.W., *Mathematics and Measurements*, London- British Museum Publications, 1987.

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