Knowledge about ancient Egyptian mathematics has been derived from a limited number of written sources, the most complete of which is the Rhind Mathematical Papyrus, written in Hieratic script by the scribe Ahmose during the Hyksos period, at about the middle of the sixteenth century BCE. The scribe was actually copying an earlier work, assembled during the Middle Kingdom, in the second half of the nineteenth century BCE, which itself was certainly drawing on earlier material, some of which probably dated back to the Old Kingdom. The document is best regarded as a teacher's or student's manual; it reveals, among other things, that the Egyptians had a counting system to base ten. Their notation for fractions, unlike ours, did not allow for numerators greater than one, except in the case of the fraction 2/3. Historians of mathematics have long presumed that such a restriction must have impeded the development of Egyptian mathematical thought. In this article, such condemnation will be rejected; instead, an account will be given of what the Egyptians were able to achieve (bearing in mind the scantiness of the surviving material, which undoubtedly fails to cover the full extent of their knowledge).
First to be considered will be the basic operations of arithmetic: addition, subtraction, multiplication, and division. If only whole numbers were involved, addition and subtraction posed no problems. Multiplication at its simplest consisted of repeated doubling. Suppose, for example, it was required to multiply some number by 37. The procedure would be to double that number and to continue doubling the result until a value was obtained for thirty-two doublings. This value, when added to the value obtained for four doublings, together with the original number, would give the answer to the multiplication. In effect, intermediate multipliers were obtained by partitioning 37 into 32 + 4 + 1. An alternative method was to take advantage of the base-ten system by introducing multiples of 10 into the calculation. Intermediate multipliers could then be obtained by partitioning 37 into 1 + 2 + 4 + 10 + 20. Partitioning numbers in this way lies at the very heart of Egyptian arithmetic.
Division was performed as a multiplication sum in reverse. If it was required to divide some number by 37, the instructions were to treat 37 so as to obtain that number. The divisor then became the number to be multiplied, and appropriate multipliers had to be found for 37, so that the products, when added together, gave the required result. Suppose the number to be divided was 47. Since 37 = 36 + 1, it would have been apparent that 37 multiplied by 1 + 1/4 gives 46 + 1/4, with a shortfall from 47 of 3/4, or 1/2 + 1/4. It follows that the remaining multiplications needed to bring the total up to 47 are 1/2 × 1/37 = 1/74 and 1/4 × 1/37 = 1/148. The answer to the division of 47 by 37 would then be written as 1 + 1/4 + 1/74 + 1/148.
The division of 47 by 37 is the same as multiplying the fraction 1/37 by 47. The Egyptians were adept at handling unit fractions, so their methods are discussed here in more detail. The addition of fractions with unit numerators (1) over different denominators was done just as one would today, by finding a common multiple for the denominators. An example is provided by problem no. 7 in the Rhind Mathematical Papyrus, where it is required to multiply 1/4 + 1/28 by 1 + 1/2 + 1/4. The working shows how to obtain the result by multiplying out and then taking 28 as a common multiple, so that the only fractions to be added at the end are halves and quarters, leading to an answer of 1/2. In fact the creator of the problem would have known quite well that 1/4 + 1/28 is equal to 2/7, since doubling of 1/7 features in a table at the beginning of the papyrus in which all odd-numbered fractions are doubled up to 1/101. He would also have known that 1 + 1/2 + 1/4 is equivalent to seven quarters (7/4), so that the answer to his sum has to be 2/7 × 7/4, or 1/2. Like many of the problems in the Rhind Mathematical Papyrus—which was designed to train scribes in the various mathematical procedures needed for their work—this one was constructed not to obtain an answer by the most direct method but to illustrate a technique.
A procedure that is fundamental to Egyptian arithmetic involves finding different series of fractions that together sum to unity (add up to 1). A number of examples occur in the Rhind Mathematical Papyrus, and it is likely that others were listed elsewhere in tables for reference. Fractional series fulfilling this condition could have been found without difficulty by selecting a number with several factors, partitioning it into components that were multiples of those factors, and then dividing by the chosen number. Suppose the chosen number was 30; it can be partitioned into 20 + 6 + 3 + 1, 20 + 5 + 3 + 2, or 15 + 10 + 3 + 2. Dividing through by 30 will give 2/3 + 1/5 + 1/10 + 1/30, 2/3 + 1/6 + 1/10 + 1/15, and 1/2 + 1/3 + 1/10 + 1/15, all of which sum to unity. These series are all used in the Rhind Mathematical Papyrus. The most formidable series of this sort arises as a corollary of problem no. 23. It consists of the eight terms 1/3 + 1/4 + 1/8 + 1/9 + 1/10 + 1/30 + 1/40 + 1/45, but it may have been obtained by putting together two four-term series: 1/3 + 1/9 + 1/30 + 1/45 and 1/4 + 1/8 + 1/10 + 1/40, each of which sums to 1/2 and, with the addition of 1/2, forms a five-term series summing to unity.
Series summing to unity play an essential role in the Rhind Mathematical Papyrus table for doubling odd-numbered unit fractions. Doubling a fraction is the same as dividing the denominator of that fraction by 2; in the case of fractions with denominators not greater than 29, work was provided to show how this was done. In all cases but one, the intermediate multipliers used in the division were taken from the fractional series obtained by repeated halving of 2/3. In the case of 2/13, the intermediate multipliers were from the series obtained by repeated halving of 1/2.
For most higher denominators, the scribe gave only the answer, together with information showing that 2, the numerator of the doubled fraction, has been in effect partitioned into 1 plus a fractional series summing to unity. For example, to double 1/67, multiply 67 successively by 1/40, 1/335, and 1/536 to get respectively 1 + 1/2 + 1/8 + 1/20, 1/5, and 1/8. The series 1/2 + 1/4 (= 2 × 1/8) + 1/5 + 1/20 sums to unity, since 20 partitions into 10 + 5 + 4 + 1, so that the addition of 1 gives the 2 of the doubling process. To take a simpler example, 2/7 = 1/4 + 1/28, because multiplying through by 7 gives 7 × 1/4 = 1 + 3/4 = 1 + 1/2 + 1/4, 7 × 1/28 = 1/4, and 1/2 + 1/4 + 1/4 sums to unity.
The evidence suggests that in the division, care has been taken to choose fractional multipliers not just from the 2/3, 1/3, 1/6 … or the 1/2, 1/4, 1/8 … series, but in such a way as to give the most elegant result (i.e., with not more than four fractions, with denominators that are if possible even numbered and not too large, so as to facilitate subsequent calculations). Trial and error with different multipliers would have been exceedingly time-consuming, but in the Robins and Shute (1987) commentary on the Rhind Mathematical Papyrus, the authors suggested a possible shortcut procedure, which would involve the partition of 2 into 1 with a series of fractions that sum to unity, as given for each example by the scribe.
Other arithmetical techniques employed by the ancient Egyptians included the summing of arithmetical and geometrical progressions, the solving of linear equations, and the use of reciprocals. The most awkward equation tackled in the Rhind Mathematical Papyrus, in problem no. 31, is (1 + 2/3 + 1/2 + 1/7)x = 33, the solution of which is equivalent to evaluating 33 × 42 ÷ 97. To lessen the burden of a long string of fractions, the scribe arranged to deal with the smaller ones separately. He found that the coefficient of x in the equation, when multiplied by 14 1/4, fell just short of 33, so he expressed that shortfall as the sum of six unit fractions.
The Egyptians obtained reciprocals by dividing numbers, integral or fractional, into unity. The reciprocal pairs of numbers that occur in the Rhind Mathematical Papyrus are 1 + 1/2 + 1/4 and 1/2 + 1/14 (problem nos. 9, 63), 4 + 1/2 and 1/6 + 1/18 (problem no. 67), and 2 + 2/3 and 1/4 + 1/8 (problem no. 71). Their modern equivalents are 7/4 and 4/7, 9/2 and 2/9, and 8/3 and 3/8. Tables of reciprocals were probably available for scribal use, but none has survived. Perhaps the main interest of reciprocals lies not in what we know the Egyptians did with them, but in what they might have done, since a single cuneiform tablet (Plimpton n.322) shows that the Babylonians knew that reciprocals could generate so-called Pythagorean triples (i.e., three whole numbers such that the square of the largest is equal to the sum of the squares of the other two). The simplest of all reciprocal pairs, 2 and 1/2, yields the simplest of all Pythagorean triples: 3,4,5. The method can be obtained by expressing the Pythagorean relationship as the difference of two squares, factorizing and dividing by the square of the middle term of the triple. Thus, 4^{2} = 5^{2} − 3^{2} yields 1 = [(1 + 1/4) + (1 − 1/4)] × [(1 + 1/4) − (1 − 1/4)] = 2 × 1/2.
Since, according to the theorem attributed to Pythagoras, Pythagorean triples form the lengths of the sides of right-angled triangles (hence their name), it is pertinent to consider whether the Egyptians knew about this property. As far as the triple 3,4,5 is concerned, it is virtually certain, despite doubts expressed by some historians of mathematics, that they did. Problem no. 1 in the Berlin Mathematical Papyrus (P. Berlin 6619) concerns the areas of three squares, of which the largest has an area of 100 square cubits, so a side is 10 cubits. Its area is equal to the sum of the areas of two smaller squares, whose sides are in the ratio of 1:1/2 + 1/4. It is required to discover the lengths of their sides. These turn out to be 8 cubits and 6 cubits; it follows that the ratio of the lengths of the sides of the three squares is 6:8:10. In other words, they can be regarded as squares based on the sides of a 3,4,5 right-angled triangle. Later, work in Demotic script in the Cairo Papyri includes sloping-pole and rectangle problems (nos. 24–31) that involve the Pythagorean triples 3,4,5; 5,12,13; and 20,21,29.
Another reason for supposing that the Egyptians knew of the 3,4,5 triangle is based on the proportions of the fourth dynasty pyramid of Khafre (Chephren) at Giza and of many of the later Old Kingdom pyramids. The same proportions occur in some pyramid problems included in the Rhind Mathematical Papyrus (nos. 56–59). They show that the eventual conformation of a pyramid was determined by the slope of its faces and the size of its base. The unit measuring slope was the seked (sḳd), giving the displacement horizontally for a vertical drop of seven units, seven being the number of palms in a royal cubit, which was the architectural unit of length. The seked for Khafre's pyramid is 5 1/4, and a lateral displacement of 5 1/4 units for a drop of 7 is the same as a lateral displacement of 3 units for a drop of 4. A vertical section through a pyramid with this seked, passing through its apex and the midpoints at two opposite sides of its base, would produce two identical 3,4,5 right-angled triangles, whose sides are half the width of the square base of the pyramid, its height, and the length from the apex to the midpoint of one side of the base. It is hard to see any reason for the change, not readily appreciated with the naked eye, from the more primitive seked of 5 1/2 occurring in the Great Pyramid, unless it was to incorporate a 3,4,5 right-angled triangle, thereby facilitating the cutting by stonemasons of casing blocks with the correct angle.
The concept of a seked can be regarded as a rudimentary form of trigonometry. Two of the greatest Egyptian achievements in mathematics belong to the sphere of geometry; they are (1) the formula, correct and by no means obvious, for the volume of a truncated pyramid that was given in the Moscow Mathematical Papyrus (no. 14) and (2) the formula (approximate as it had to be, but still the best in the pre-Hellenic world) for the area of a circle that was given in the Rhind Mathematical Papyrus (nos. 41–43, 48, and 50). The methods for arriving at these formulas are not extant; nor is it anywhere attested that the Egyptians knew the formula for the volume of a complete pyramid—but they surely must have, since to compute it they had only to reduce the top surface of a truncated pyramid to zero.
The Egyptian method for estimating the area of a circle was to subtract a ninth part from its diameter and square the result, with an error of only 0.6 percent. Attempts have been made to explain the way this formula was obtained by considering problem no. 48 in the Rhind Mathematical Papyrus. This problem is unusual because there is no description of the procedure but simply a calculation accompanied by a diagram, which has variously been interpreted as either a circle or an octagon inscribed within a square. The square is given a side of 9 units, so that if the inscribed figure is a circle, it will have a diameter of 9 units, and its area, according to the Egyptian formula will be 64 square units, as compared with the 81 square units making up the circumscribed square. If the inscribed figure is an octagon, the intention may have been to treat it like a circle, thereby getting an approximate value for its area. If the octagon was meant to have angles located at trisection points along the sides of a circumscribed square with sides of 9 units, its area would be equal to that of five small squares, each with sides of 3 units, plus four triangular half squares together equal to two whole squares also with sides of 3 units, giving an area of 7 × 9 = 63 square units. This is close to the area of a square with sides of 8 units or to that of a circle with a diameter of 9 units, according to the Egyptian formula.
Against the above interpretation, it seems evident that the Egyptian scribe was more concerned to prove his methods by showing that they worked than to indicate how those methods were reached. An alternative hypothesis would suppose that the circle whose area was to be determined was not inscribed within a square but super-imposed upon it, being drawn through quarter points along its sides. Such a circle and square would be seen to have approximately equal areas, so that the circle would have been effectively “squared.” The relative proportions of the diameter of the circle and the side of the square would be found to be very close to 9:8, as in the Egyptian formula for the area of a circle.
From the discussion above, the competency of Egyptian mathematicians would seem established. Comparisons with Babylonian achievements seem adverse because so much more Babylonian material has survived on their clay tablets than on fragile Egyptian papyrus. If the Moscow Mathematical Papyrus had not survived, no record would exist of the Egyptians' ability to calculate the volume of a truncated pyramid. Other such procedures that were well known to the Egyptians may have left no traces in the record known so far. Then, too, the supposed impediment caused by the use of unit fractions is largely illusory (since these were a notational device, perhaps adopted in part for aesthetic reasons), and not the result of a conceptual block. The ability to manipulate such fractions deftly may have been a source of pride to the Egyptian scribe.
Bibliography
- Chase, Arnold B. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Transcriptions and Literal Translations. Classics in Mathematics Education, volume 8, 1979. A good overview, convenient for the general reader, produced by an enthusiast.
- Gillings, Richard J. Mathematics in the Time of the Pharaohs. Toronto, 1972; Dover reprint, 1982. A highly readable, personal approach; the Berlin and Moscow problems referred to were transcribed on pp. 161 and 188.
- Parker, Richard A. Demotic Mathematical Papyri. Providence, 1972. A brief commentary and seventy-two texts from Cairo, the British Museum, and Carlsberg (not all free from Babylonian and Greek influences); the Demotic problems referred to were included.
- Peet, Thomas E. The Rhind Mathematical Papyrus BM 10057 and 10058: Introduction, Transcription, Translation and Commentary. Liverpool and London, 1923. The definitive publication, the first in English. Reviewed by B. Gunn, Journal of Egyptian Archaeology 12 (1926), 123–137.
- Peet, Thomas E. “Mathematics in Ancient Egypt.” Bulletin of the John Rylands Library 15 (1931), 404–441. A straightforward general account by an Egyptologist trained in mathematics.
- Robins, Gay. “Mathematics, Astronomy and Calendars in Pharaonic Egypt.” In Civilizations of the Ancient Near East, edited by J. M. Sasson, vol. 3, pp. 1799–1813. New York, 1995. An up-to-date survey.
- Robins, Gay, and Charles C. D. Shute. “Mathematical Bases of Ancient Egyptian Architecture and Graphic Art.” Historia Mathematica 12 (1985), 107–122. Includes a study of pyramid proportions and a warning against mathematical coincidences.
- Robins, Gay, and Charles C. D. Shute. The Rhind Mathematical Papyrus. London, 1987, Dover reprint, n.d. Includes color photographs of the recto and verso of BM 10058 and the recto of BM 10057, constituting all the purely mathematical part of the papyrus; the commentary covers selected problems and gives a detailed analysis of the table of doubled odd-numbered unit fractions.
- Robins, Gay, and Charles C. D. Shute. “The 14:11 Proportion in Egyptian Architecture.” Discussions in Egyptology 16 (1990), 75–79. The authors reject the notion that the 14:11 proportion found in the Great Pyramid was chosen to form part of a geometric series with that common ratio.
- Robins, Gay, and Charles C. D. Shute. “Irrational Numbers and Pyramids.” Discussions in Egyptology 18 (1990), 43–53. Despite claims by modern numerologists, irrational numbers such as pi and phi occur in the dimensions of the Great Pyramid only as mathematical coincidences.
Charles Shute