Mathematics

Sources

Mathematical Knowledge. The Egyptians developed practical knowledge of arithmetic, algebra, geometry, and trigonometry. They applied these mathematical operations to practical problems such as figuring out the number of people needed to dig a lake of a particular size, build a ramp, transport an obelisk, erect a colossus, or provide necessities for an army. They could also compute the area of buildings and the slope of a pyramid. The most important mathematical text preserved from ancient Egypt is the Rhind Mathematical Papyrus, now located in the British Museum. Some fragments of the papyrus are also kept in the Brooklyn Museum of Art. A scribe named Ahmose recopied this text in the Second Intermediate Period (1630-1539/1523 b.c.e.) from a copy he claimed was made in the reign of Amenemhet III (circa 1818-1772 b.c.e.). Many commentators believe the text also draws on materials already known in the Old Kingdom (circa 2675-2130 b.c.e.). Nevertheless, this text probably does not contain all mathematical knowledge available to ancient Egyptians. Egyptian mathematics was sophisticated enough that major engineering feats, such as the Great Pyramid and other monumental buildings, could be constructed.

Arithmetic and Algebra. Arithmetic is the manipulation of numbers, including fractions. Algebra uses arithmetical operations to solve an equation for an unknown number. Modern arithmetic is taught as adding, subtracting, multiplying, and dividing numbers. These operations are also used to solve equations. The basic Egyptian method was doubling; their number system was base 10, similar to the modern system of numbers. The doubling method is best illustrated in the following solution to the problem 47 x 33:

Here the number 47 is doubled, that sum is doubled, then that sum is doubled, until the result comes within one

unit of the desired answer (that is, 32 x 47). Then the one additional unit of 47 can be added to result in the answer, 33 x 47=1551. Using a set of tables reproduced in the Rhind Mathematical Papyrus, a similar set of operations could be performed with fractions. This table allowed the Egyptians to solve for unknowns (algebra).

Geometry . Geometry is the measurement of areas and volumes. Egyptian accomplishments in geometry include knowledge of ? and of the special properties of right-angle triangles. They knew the accurate value of ? to within 0.6 percent—3.16 rather than 3.1416, as used for modern calculations. This value is needed to calculate the area of a circle. It represents a ratio between the circumference of a circle and its diameter. Using ?, the Egyptians could calculate both areas and volumes. The Egyptians also understood the special properties of the right-angle triangle, which include the proportions of the sides (3:4:5) and the fact that the square of the sides is equal to the square of the hypotenuse (the side opposite the ninety-degree angle). These calculations made it possible to square the corners of the pyramids.

PROBLEM 57 IN THE RHIND MATHEMATICAL PAPYRUS

The following problem describes how to compute the height of a pyramid.

A pyramid 140 (cubits) in length of side, and 5 palms and a finger (1/4 palm) is its slope (seked). What is the vertical height of it?

You are to divide one cubit by the slope doubled, which amounts to 10/12; You are to reckon with 10½ to find 7 for this is one cubit.

Reckon with 10½: two-thirds of 10½ is 7.

You are now to reckon with 140 for this is the length of a side:

Make two-thirds of 140, namely 93 (cubits). This is the vertical height of it.

Source: Gay Robins, “Mathematics, Astronomy, and Calendars in Pharaonic Egypt,” in Civilizations of the Ancient Near East, volume 3, edited by Jack M. Sasson (New York: Scribners, 1995), p. 1810.

Trigonometry . The Egyptians needed to calculate the slope of a pyramid in order to know that the four sides would meet in a point at the top. They also had to calculate the height of a pyramid given a particular length of a side and a particular slope. The slope was called the seked, the lateral displacement for a drop of one unit. The primary seked for an Old Kingdom pyramid was 5.5. The Rhind Mathematical Papyrus used a seked of 5.5 in its problems, suggesting that the text used materials dating from the Old Kingdom.

Sources

Arnold Buffum Chace, The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Transcriptions, Transliterations, and Literal Translations (Reston, Va: National Council of Teachers of Mathematics, 1979).

Richard J. Gillings, Mathematics in the Time of the Pharaohs (Cambridge, Mass.: MIT Press, 1972).

Gay Robins and Charles Shute, The Rhind Mathematical Papyrus: An Ancient Egyptian Text (London: British Museum Publications, 1987).