Skeptic, Vol. 5, No. 4, 1997
Reproduced with permission
The Great Pyramid of Giza, dating from the 26th century BCE, is one of the most magnificent edifices on Earth. Its sides measure 755.43 ft., 756.08 ft., 755.88 ft., and 755.77 ft. at the base, and its original height was 481.4 ft. (Edwards, 1961, p. 118). As originally designed in ancient units of measure, it was to have been 440 cubits on a side and 280 cubits high (Wheeler, 1935, pp. 15, 183). The 280cubit height was probably derived from the particular unit of measure used. The Egyptian “royal cubit” was subdivided into 7 handbreadths, and each handbreadth into 4 fingerbreadths, for a total of 28 fingerbreadths per cubit. It is not at all difficult to imagine an ancient architect standing his cubit stick on end and envisioning a scale of 10 cubits per each fingerbreadth for the height of the proposed structure.
As many commentators in the last century and a half have noted, the dimensions of the Great Pyramid exhibit an interesting mathematical relationship. The ratio of its height to its perimeter is the same as that of a circle’s radius to its circumference. If the square base of the Great Pyramid were made into a gigantic circle having the same perimeter, the diameter of that circle would equal twice the height of the pyramid. Dividing the circumference of a circle by its diameter gives that special value called pi (p), 3.1415926…, usually rounded to 3.14. Calculations based on the actual measurements of the Great Pyramid given above in decimalized feet yield the extremely close value of 3.1399667; calculations based on the original measurements in cubits yield the even closer value of 3.1428571, exactly 3^{1}/_{7}.
What are we to make of this pi in the pyramid? To Afrocentrists, it is yet another indication of the mathematical sophistication of “black” Egyptians. It is also seemingly a vindication of the “stolen legacy” thesis, because Archimedes, the great Greek scientist of the 3rd century BCE, is the one always associated with the 3^{1}/_{7} approximation of pi. Archimedes must have stolen pi from the Egyptians, and thus the blacks.
How could the Egyptians have discovered the 3^{1}/_{7} value for pi 23 centuries before Archimedes? A very plausible explanation would be that they could have discovered a 3^{1}/_{7} circumference to diameter ratio by simply measuring round objects. As already noted, the Egyptian royal cubit was divided into 7 handbreadths. A round object 1 cubit (7 handbreadths) in diameter would have a measured circumference of 3^{1}/_{7} cubits (22 handbreadths).
As reasonable as this might seem, however, there is absolutely no evidence that the Egyptians ever used these measurements to establish a circumference to diameter ratio. Had they done so, we would expect to find evidence of pi = 3^{1}/_{7} in subsequent mathematical texts dealing with circle problems. Instead, we find evidence for less accurate values of pi and, more disturbingly, evidence that the ancient Egyptians might not have actually understood the significance of the concept of pi.
If the ancient Egyptians weren’t clear on the concept of pi, though, then how do we explain the dimensions of the Great Pyramid? Actually, such a configuration could have been developed quite simply and naturally, without its designers ever being aware of the mathematical significance of its proportions.
Let’s say that some prepyramid Egyptian peasant wants to construct a small, round, mud brick hut with a conical thatched roof. He would first take a couple of sharpened stakes, connect them with a length of cord, and pound one of the stakes into the ground. He would then pull the cord taut and use the pointed end of the free stake to inscribe a circle on the ground to mark the outside circumference of the hut. After building the hut, he would then lay out another circle of the same size and use it as a guide to “prefab” the conical roof frame, which would then be lifted into place atop the hut. How high should he make this roof frame? One value that immediately suggests itself is the length of the radius of the circle. All he would have to do is lay a long stick down with one end at the center of the circle (the hole left by the stake) and then mark the spot near the other end where the stick extends across the circle. He could then plant the stick upright in the stake hole so that only the measured off portion equal to the radius would be above ground. This would allow him to calculate the length of the “rafters” by marking off on another long stick the distance between the top of the planted stick and the periphery of the circle.
Now let’s say that the hut builder wants to build a second hut, the same size as the first, but square instead of round. The builder isn’t mathematically sophisticated enough to calculate floor space, so he makes the square hut “equal” to the round one by giving it the same perimeter. All he would have to do is stretch a cord around the round hut to measure its circumference, then fold the cord into quarters to get the appropriate wall length of the square hut. The roof frame of the square hut would be a “square cone” – a pyramid – having a perimeter equal to the circumference of the conical roof frame. Since the builder wants to keep the round and square huts otherwise as similar as possible, he would give the pyramidical roof frame the same height as the conical roof frame – a height equal to the radius of the conical frame.
The pyramidical roof frame just described – which could have easily been conceived and built by a mathematically unsophisticated peasant with absolutely no knowledge of pi – has exactly the same proportions as the Great Pyramid. This can be dramatically illustrated by considering the dimensions of a round hut seven cubits in diameter (this measurement is derived from the scale of seven handbreadths to the cubit). Its 3^{1}/_{2} cubit radius is also the height of its roof frame and the height of the pyramidical roof frame; ^{1}/_{4} of its circumference gives the length of the sides of the square hut and the pyramidical roof frame. Progressively doubling these dimensions gives the following values:
Multiplying the values in the last row by 10 gives the following measurements of the Great Pyramid in cubits: height, 280; doubled height, 560; perimeter, 1760; and length of sides, 440.
Let’s now turn our attention to actual mathematical texts from ancient Egypt (see Gillings, 1982). The Rhind Mathematical Papyrus (RMP) dates from the 17th century BCE, but was copied from a lost original probably dating from the 19th century BCE. Problem 50 gives the procedure for finding the area of a circle of a given diameter; it boils down to the formula A = (^{8}/_{9}D)^{2}. Problem 41 deals with the volume of a cylinder, calculated by multiplying the area of the circular base by the height: V = (^{8}/_{9}D)^{2}H. Problem 43 is a shortcut method for finding the volume of a cylinder in a particular unit of capacity. Although it does not explicitly mention squaring ^{8}/_{9} of the diameter, it is based upon that method and gives exactly the same answer. This shortcut method also shows up in the fragments of the Kahun Papyrus (19th century BCE), indicating that its author was also familiar with A = (^{8}/_{9}D)^{2}. Problem 10 of the Moscow Mathematical Papyrus (19th century BCE) deals with finding the area of a hemispherical basket of a given diameter. The method given boils down to the formula A = 2(^{8}/_{9}D)^{2}, the hemisphere having twice the area of a circle of the same diameter.
The area of a circle formula employed in these papyri was probably derived in the following way (Gillings, 1982, pp. 139146): The Egyptians found that a circle inscribed inside a square had approximately the same area as a semiregular octagon formed by cutting off the corners of the square. The known area of the octagon could then be approximately expressed in the form of another square. A square of 9 units on a side has an area of 81 square units. Removing the comers – 45° right triangles measuring 3 units on their like sides – leaves a semiregular octagon of 63 square units. This semiregular octagon has approximately the same area as a square of 8 units on a side, which has an area of 64 square units. In other words, a circle 9 units wide has approximately the same area as a square 8 units wide.
We can extrapolate a value for pi from this Egyptian formula by rearranging our own formula for the area of a circle, A = pr^{2}, into p = A/r^{2}, plugging in the dimensions given above, D = 9 and A = 64, and solving for p: p = 64/(^{9}/_{2})^{2} = 64/20.25 = 3.1604938, usually rounded to 3.16 or expressed in fractional form as 3^{13}/_{81}. It is readily apparent that this value for p is less accurate than the 3.1428571 (3^{1}/_{7}) derived from the measurements of the Great Pyramid. If the Egyptians had known that a circle’s circumference to diameter ratio was 3^{1}/_{7} when they built the Great Pyramid, why isn’t there evidence of that value in these subsequent mathematical papyri dealing with areas of circles? Why do they all instead yield the same less accurate value? The only logical explanation is that the pyramid builders weren’t even aware of the significance of the relationship between the Great Pyramid’s height and perimeter.
A final irony that illustrates the Egyptian obliviousness to 3^{1}/_{7} as a value of pi shows up in the “fraction of a hekat” problems in the RMP (Gillings, 1982, pp. 205206). In these problems, different sized graduated measuring scoops are used to fill a container that holds exactly 1 hekat (about ^{1}/_{8} bushel). The problems involve calculating the capacity of the measuring scoop as a fractional part of the hekat. Problem 38 states that 3^{1}/_{7} scoops are required to measure out 1 hekat. One hekat divided by 3^{1}/_{7} gives ^{7}/_{22} hekat as the capacity of the scoop. The calculations here – dividing 1 by 3^{1}/_{7} – could have just as easily been used to calculate the diameter of a circle 1 unit in circumference had the Egyptians been aware of the significance of 3^{1}/_{7} as a circumference to diameter ratio. Amazingly, they had a very accurate approximation of pi right under their noses and weren’t even aware of it.
If we grant that the Egyptians missed the boat as far as pi = 3^{1}/_{7} is concerned, can’t we at least give them credit for discovering that pi =3^{13}/_{81} (3.16)? Although many sources do indeed cite these figures as “the Egyptian value for pi,” they are carelessly misleading.
The whole concept of pi involves the recognition that dividing a circle’s circumference by its diameter yields a special constant number – pi. – that can in turn be used to solve a variety of circle problems. Dividing the known circumference of any circle by pi will give the unknown diameter; multiplying a known diameter by pi will give the unknown circumference; squaring the known radius of a circle and multiplying by pi will give the unknown area; and so on. The simple truth is that there are absolutely no problems in any of the extant prehellenistic Egyptian mathematical texts that deal with calculating the diameter or circumference of a circle or the lateral area of a cylinder of a given diameter – the very problems that would require some kind of circumference to diameter ratio. All we have are procedures for finding the area of a circle, the area of a hemisphere, and the volume of a cylinder, all based on A=(^{8}/_{9}D)^{2}, a formula that relates diameter to area but not to circumference.
Furthermore, in order to derive this socalled “Egyptian value of pi,” we had to plug a given diameter and area into our own formula, A = pr^{2}, which already presupposes that special relationship between a circles diameter and circumference known as pi. In doing this, we retroactively smuggled pi into Egyptian mathematics, where it had not previously existed.
Although reading pi into these older prehellenistic papyri was a mistake, pi does in fact show up much later in a demotic papyrus of the 3rd century BCE, P Cairo J. E. 8912730, 8913743 (Parker, 1972, pp. 4041). Problems 32 and 33 deal with finding the diameter of a circle of a given area. The procedure involves calculating the square root of 1^{1}/_{3} the area: D = (A + ^{1}/_{3}A). The principle involved here is the assumption that a circle has ^{3}/_{4} the area of its circumscribing square. Increasing the circle’s area by ^{1}/_{3} therefore gives it the same area as the square; taking the square root of this amount gives the length of the side of the square, which is equal to the diameter of the circle. The answers to the problems are then checked by the following procedure described in the text: First, the diameter is multiplied by 3 to obtain the circumference. This is an explicit use of 3 for pi. The circumference is then divided by 4. Finally, the result of this division, ^{1}/_{4} circumference, is multiplied by ^{1}/_{3} circumference = (diameter) to get back to the original area: A = ^{1}/_{4}C x ^{1}/_{3}C.
So here we finally have the first attested use of a standalone constant for pi in Egyptian mathematics – but it is a decidedly less accurate value than the 3^{13}/_{81} value implicit in the older prehellenistic papyri. Furthermore, 3 as a value of pi and the circle to square area ratio of ^{3}/_{4} used to solve the problems above are not even Egyptian – they’re taken straight out of Babylonian mathematics. The A = ^{1}/_{4}C x ^{1}/_{3}C formula cited above can also be written as A= ^{1}/_{12}C^{2}, a well attested Babylonian formula (Bunt, Jones, and Bedient, 1988, p. 62). If the Egyptians had ever possessed a more accurate explicit value for pi of their own, why would they have abandoned it for an inferior Babylonian value? (There is a certain irony about this. Extreme Afrocentrists are always accusing the Greeks of stealing from the Egyptians; yet here we have the Egyptians “stealing” from the Babylonians!)
In spite of the inaccuracy of the Babylonian value of 3 for pi, it was nonetheless part of a system which did accurately describe the relationship between squares and their inscribed circles. In this relationship, the ratio of the circumference of the circle to the perimeter of the square is equal to the ratio of the area of the circle to the area of the square. A circle 1 unit in diameter would have a circumference of 3 units, which would be ^{3}/_{4} the perimeter of a square 1 unit on a side; the area of the circle would also be ^{3}/_{4} the area of the square. If the circumference and area of the inscribed 1 unit diameter circle are calculated using any other value for pi, the circle values will both be some identical fractional part of the corresponding square values.
If the Egyptians really were the great theoretical mathematicians that extreme Afrocentrists make them out to be, why didn’t they pick up on this? They had their own circle to square area ratio of ^{64}/_{81}, based on the formula A = (^{8}/_{9}D)^{2}. A square 9 units on a side (= a perimeter of 36) would have an area of 81 square units. The inscribed circle, 9 units in diameter, would have an area of 64 units. Multiplying the 36unit perimeter of the square by ^{64}/_{81} would give the circumference of the circle, 28^{4}/_{9}. Dividing this circumference by the 9unit diameter would give the explicit value of 3^{13}/_{81}, for pi. Instead of doing this, however, the Egyptians uncritically accepted the much less accurate value of 3 as given by the Babylonians. The Egyptians didn’t even attempt to revamp the Babylonian method for finding the diameter of a circle of a given area, which would have been a very simple procedure. All they would have had to do was calculate the difference in area between a square and an inscribed circle. In the case above, where the square was equal to 81 and the circle to 64, the difference is 17. Therefore, ^{17}/_{64} would be the amount by which a circle’s area would have to be increased so that its square root would equal the diameter of the circle: D = √ (A + ^{17}/_{64}A). This would have given a much more accurate value than the Babylonian version of the formula. Why didn’t the Egyptians adapt the valid Babylonian insights to their own mathematical methods, which would have enabled them to surpass the Babylonians in accuracy of calculation? Why did they merely copy the old Babylonian methods instead? It must be noted, too, that this copying took place during the same time frame when Archimedes and other Greeks associated with Alexandria were allegedly plagiarizing Egyptian mathematical knowledge.
Let’s now take a look at the works of Archimedes himself (Heath, 1953, pp. 9398). Proposition 3 of Measurement of a Circle states: “The ratio of any circle to its diameter is less than 3^{1}/_{7} but greater than 3^{10}/_{71}.” The method Archimedes used to demonstrate this can best be understood by visualizing a circle with one hexagon inscribed on the inside and another hexagon circumscribed around the outside. It is obvious that the circumference of the circle is less than the perimeter of the outer hexagon and greater than the perimeter of the inner hexagon. It is also obvious that the more sides the inner and outer polygons have, the more closely they will approximate the circumference of the circle sandwiched between them. The perimeters of the polygons, because they are made up of straight lines, can be calculated very accurately. Archimedes kept doubling the number of sides on his polygons until he came up with 96 sided figures. The outer polygon had a perimeter 3^{1}/_{7} times the diameter of the circle; the inner polygon had a perimeter 3^{10}/_{71} times the diameter of the circle.
There’s nothing comparable to this meticulous measurement of a circle’s circumference in any extant Egyptian mathematical work. In fact, the Cairo papyrus with its Babylonian borrowings appears to be the only Egyptian mathematical text that even mentions circumference. Archimedes also goes one up on the Great Pyramid by proving that pi is actually somewhat smaller than 3^{1}/_{7}. Although subsequent mathematicians refined Archimedes’ method by using polygons of increasingly more sides, the actual method itself was not challenged for over 1800 years (Beckmann, 1993, p. 110111).
Although Egyptian civilization long antedated Greek civilization and undoubtedly influenced it to some degree, Afrocentrists err when they maintain that everything the Greeks knew either had to have been learned from or stolen from the Egyptians. In the 7th century BCE, the Egyptians allowed the Greeks to found the trading port of Naucratis in the Nile delta. This would have given the Greeks plenty of exposure to Egyptian ideas. Alexandria, another Nile delta city, was founded by Alexander the Great over three centuries later. This means that the Greeks had over three hundred years before the founding of Alexandria, not only to learn whatever the Egyptians might have had to offer, but also to develop their own mathematical tradition as well. The Greeks associated with Alexandria – including Archimedes – simply didn’t need to steal anything from the Egyptians; by this time, the former students had for the most part far surpassed their former teachers. This was especially so insofar as the calculation of pi was concerned.
APPENDIX: NOT MEASURING UP
Afrocentrist Beatrice Lumpkin, Associate Professor of Mathematics at Malcolm X College in Chicago, has this to say about the measurements of the Great Pyramid (1983, p. 71):
…The square base originally measured 755.43 ft., 755.88 ft., and 755.77 ft. on its north, south, east and west sides respectively. The height was 481.4 ft….It is a fact that half the perimeter of the base divided by the height is 3.1408392; compared to the modern value of pi of 3.1415927 the difference is only 0.0007535. This is dismissed as mere coincidence by the official Egyptologists who point out that the Egyptian value for pi was, 4 x (^{8}/_{9})^{2} = 3.1604938, a difference of 0.0189012 from the modern value. Either way it is taken, it is clear that the Egyptian value for pi was more accurate than any other approximation of ancient time. I would suggest that we have too few mathematical documents at this time to reject, out of hand, the possibility that the ratio of the Great Pyramid semiperimeter to its heights was significant.
There are a number of problems with this description. First of all, Lumpkin has left out one of the four base measurements, the missing dimension being 756.08 ft. (Edwards, 1961, p. 118). When we add the four base measurements, we come up with a perimeter of 3023.16. Dividing half this perimeter, 1511.58, by the height of 481.4 gives a figure of 3.1399667 instead of the 3.1408392 cited in the text.
Lumpkin came up with her figure by rounding the 1511.58 semiperimeter up to an even 1512 and dividing by the unrounded height of 481.4. Since she is carrying her calculation of pi way out to the seventh decimal place, however, she should be using the exact values given and not be doing any rounding at all. (Rounding off the measurement defeats the very purpose of a fastidiously accurate pi calculation – something that a professor of mathematics ought to know.)
Just why did she round off the semiperimeter? Well, if she had rounded off the height as well, she would have gotten a value of 3.1434511. If she had rounded off the whole perimeter to 3023, taken half of this, 1511.5, and divided by the unrounded height of 481.4, she would have gotten 3.1398005; had she divided by the rounded height of 481, she would have gotten 3.1424116. Of all these wrong ways to go about calculating an 8digit value of pi from the measurements of the Great Pyramid, Lumpkin’s gives the closest value to the actual value of pi. In other words, she has manipulated the data in order to make the Great Pyramid value of pi appear to be more accurate than it really is, which, of course, makes this supposed achievement of the ancient Egyptians appear all the more impressive.
Lumpkin also doesn’t seem to be aware that the differences in the lengths of the sides of the Great Pyramid indicate that there were minor construction misalignments that could throw off the value of pi as derived from the actual measurements. This metrological problem could have been circumvented by considering the intended dimensions of the structure in their original units of measure, the royal cubit. The height of the Great Pyramid was to have been 280 cubits and each of its sides 440 cubits (Wheeler, 1935, pp. 15, 183). Dividing 280 into the 880 cubit semiperimeter gives 3.1428571, exactly 3^{1}/_{7}. At no point in her article, however, does Lumpkin discuss the dimensions of the Great Pyramid in cubits.
Lumpkin also mentions another derived Egyptian value for pi, 3.1604938, and seems totally unaware of a major problem in the chronology of these values. The Great Pyramid value of 3.14 dates from the 26th century BCE. The 3.16 value comes from papyri dating from the 19th17th century BCE. If the 3.14 figure was mathematically “significant,” and could perhaps be proven so by yet to be discovered texts as Lumpkin suggests, then why did it disappear, only to be replaced by a less accurate value centuries later? How could the Egyptians simply have lost a significant mathematical value that they had conspicuously incorporated into the greatest edifice ever erected by humankind? Nearly everyone in Egypt must have heard of the Great Pyramid, 280 cubits high and 440 cubits on a side.
Finally, Lumpkin states that either of the two derived values of pi cited above was more accurate than any other in antiquity This is simply not true. Although the Babylonians used 3 as a handy value for pi, they were aware of its shortcomings and developed a “constant” or “coefficient” ‘to correct these shortcomings when greater accuracy was desired. After calculating the area of a circle using their formula A = ^{1}/_{12}C^{2} (which is the same as A = C^{2}/4p where p = 3), they would multiply the result by ^{24}/_{25} (= .96) to obtain the more accurate value. This more accurate value was exactly the same as would have been obtained had 3^{1}/_{8} (3.125) been used for pi (Bruins and Rutten, 1961, pp. 18, 33) – and 3.125 is closer to the actual value of pi than 3.1604938.
BIBLIOGRAPHY
Beckmann, P. 1993. A History of p (pi). New York: Barnes & Noble Books.
Bruins, E. M. and M. Rutten. 1961. Textes Mathématiques de Suse. Paris: Librairie Orientaliste Paul Geuthner.
Bunt, L. N. H., P. S. Jones, and J. D. Bedient. 1988. The Historical Roots of Elementary Mathematics. New York: Dover.
Edwards, I. E. S. 1961. The Pyramids of Egypt. Baltimore: Penguin Books.
Gillings, R. J. 1982. Mathematics in the time of the Pharaohs. New York: Dover.
Heath, T L. (ed.) 1953. The Works of Archimedes. New York: Dover.
Lumpkin, B. 1983. “The Pyramids: Ancient Showcase of African Science and Technology,” in Blacks in Science. Ancient and Modern. Van Sertima (ed.), New Brunswick: Transaction Books.
Parker, R. A. 1972. Demotic Mathematical Papyri. Providence: Brown University Press.
Wheeler, N. F. 1935. ” Pyramids and Their Purpose.” Antiquity. 9:521. ______. 1935. “Pyramids and Their Purpose. II: The Pyramid of Khufu..” Antiquity. 9:161189.
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