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Did Egyptian Scribes Have an Algorithmic Means for Determining the Circumference of a Circle?

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Did Egyptian Scribes Have an Algorithmic Means for Determining the Circumference of a Circle? View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematica 38 (2011) 455–484 www.elsevier.com/locate/yhmat Did Egyptian scribes have an algorithmic means for determining the circumference of a circle? Leon Cooper * 540 Westminster Rd., Putney, VT 05346, USA Available online 20 July 2011 Abstract It has been at various times proposed in regard to Problem 10 of the Moscow Mathematical Papyrus that Egyptian scribes had developed a computational algorithm by which they could calculate the circumference of a circle of known diameter length. It has also been proposed at various times that the 7:22 ratio between a circle’s diameter and its circumference was known by the Egyptian scribes at a surprisingly early period. The present paper explores currently available evidence that lends support to each of these propositions. Ó 2011 Elsevier Inc. All rights reserved. Zusammenfassung Es ist verschiedentlich angenommen worden, dass die ägyptischen Schreiber für das Problem 10 des Moskauer Papyrus einen Algorithmus entwickelt haben, mit dem sie den Umfang eines Kreises bei gegebenem Durchmesser berechnen konnten. Es ist ebenfalls verschiedentlich behauptet worden, dass das 7:22 Verhältnis zwischen dem Kreisdurchmesser und dem Kreisumfang den ägyptischen Schreibern schon überraschend früh bekannt gewesen ist. Der vorliegende Artikel untersucht gegenwärtig vorhandene Evidenz, die diese Annahmen stützt. Ó 2011 Elsevier Inc. All rights reserved. MSC: 01A16; 14-03 Keywords: Moscow papyrus; Rhind papyrus; Egyptian mathematics; Circumference; Circle; 7:22 1. Introduction Only a handful of known surviving ancient Egyptian mathematical texts can be dated to as early as the Middle Kingdom Period (ca. 2119–1794 B.C.), and in one of these, the Rhind * Fax: +1 802 387 2120. E-mail address: [email protected] 0315-0860/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.hm.2011.06.001 456 L. Cooper Mathematical Papyrus, we learn that Egyptian scribes had at their disposal a computa- tional algorithm for finding the area of a circle of known diameter.1 The method of this algorithm was to subtract a 1/9th part from a circle’s diameter, and then to multiply the remaining length by itself — that is, to “square” 8/9ths of the diameter’s length. In modern terms this would be written as: circle’s Area = (8/9 Â D)2. It is worth noting that the method used by the scribes is both easy to remember and easy to implement. In addition, it provides a remarkably accurate way in which to determine a circle’s area, with the answer that it gives being too large by only about 3/5 of 1%. Many theories have been advanced on how the scribes came to discover this approach. For the moment, we will simply acknowledge that the discovery was a significant achieve- ment, and that it indicates not only a strong desire to possess such a capability, but also the sophistication and ingenuity to devise so workmanlike a procedure. It may therefore seem a little surprising that these same extant mathematical texts do not include problems that definitively reveal that the scribes had also developed a computational method by which to determine the circumference of a circle of known diameter. One might argue that there was not as great a need for a circumference algorithm, since it was the computation of grain volumes housed in cylindrical structures that was apparently of greatest importance at the time — and as we shall see, to do this all the scribe needed to know was the height of the cylindrical granary and its diameter. However, for any artisan or builder who is creating a circular construct, it can often be more helpful to know the length of the arc one needs to fashion rather than the area of the circle that may be involved. To this can be added that fashioning a means by which to accurately compute circular area is an inherently more complicated undertaking than arriving at an algorithm for find- ing a circle’s circumference — since this latter is easily derivable from a simple empirical measurement of a circle’s circumference relative to its diameter. Furthermore, the circum- ference undertaking becomes even easier when one is using a measurement unit that is con- venient to the situation at hand — and as it happens, precisely such a measurement unit was indeed in constant use in ancient Egypt. All of these particulars will be explored in more detail throughout the course of this essay. For now, let us begin with a discussion of Problem 10 of the Moscow Mathematical Papyrus. 2. Problem 10 of the Moscow Mathematical Papyrus (MMP 10) Of all the problems in the extant ancient Egyptian mathematical papyri, it is Problem 10 of the Moscow Mathematical Papyrus that offers the strongest suggestion that the scribes of the Middle Kingdom period (ca. 2119–1794 B.C.) had devised a procedure for computing the circumference of a circle. Although there are differing interpretations regarding the intended meaning of some of the terms that appear in MMP 10, there is fairly uniform agreement regarding the specific individual steps that are taken by the problem’s central computational regime. 1 Although the writing of the Rhind Mathematical Papyrus has been dated to the Second Intermediate Period (ca. 1794–1550 B.C.), much of the work in it is clearly based on earlier scribal text material. The other two mathematical papyri to be considered in this essay date to the Middle Kingdom period. See Imhausen [2003, 15] for a listing of known ancient Egyptian mathematical papyri and their dates. Ancient Egyptian Scribal Circumference Algorithm 457 For the purposes of this analysis, we will focus our attentions on the interpretations of MMP 10 put forward by Peet [1931a], Hoffmann [1996], and the present author [Cooper, 2010]. We will start with the second of the two interpretations proposed by Peet [1931a, 103–106]. Due to damage to the papyrus, there is a word that is all but completely missing from the problem’s sixth line. Peet [1931a, 105] proposed that this missing word was ipt, an ancient Egyptian term at times used to denote a cylindrical grain container. He then reasoned that the word nb.t, which appears in the problem’s first and fifth lines, would have been intended to mean a “semicylinder.” His suggested translation of the problem was, therefore, 1. Example of working out a semicylinder (nb.t) 2. If they say to you a semicylinder (nb.t) of 4 1/2 in mouth (diameter) 3. by 4 1/2 in height (i.e., the proposed semicylinder’s length) 4. Let me know its surface area. You are to 5. Take a ninth of 9, since a semicylinder (nb.t) 6. is half of a cylinder (ipt); result 1 7. Take the remainder, namely 8 8. Take a ninth of 8 9. result 2/3 + 1/6 + 1/18. Take 10. the remainder of these 8 after subtracting 11. the 2/3 + 1/6 + 1/18; result 7 1/9 12. Reckon 7 1/9 4 1/2 times 13. Result 32. Behold, this is its (surface) area 14. You will find it correct. Hoffmann’s interpretation of this problem closely followed that shown above, with the only major difference being that he proposed that the nb.t term was likely intended to refer to a semicylindrically shaped vaulted ceiling, such as are found in certain ancient Egyptian tombs. The present author’s interpretation also accepted that the damaged word in line 6 was intended to be the word ipt, and so agreed that lines 5 through 11 essentially relate to the computation of the circumference of a circular object. If any one of these three interpretations is correct, this would then mean that lines 5 through 11 represent, in essence, the workings of an algorithm designed to computationally determine the circumference of a circle of a given diameter. These interpretations propose that since what is apparently being sought in this particular problem is only one-half of the circumference, the calculations have the diameter being multiplied by a factor of 2, with the clear implication that for a full circumference computation the diameter would need to be multiplied by a factor of 4. The method implied by MMP 10 for a complete circumference would therefore be A. Multiply the length of the circle’s diameter by 4 (that is, find 4 Â D). B. Find the 1/9 part of this 4 Â D amount and subtract it from 4 Â D (that is, find 8/9 of 4 Â D). C. Find the 1/9 part from this new amount and subtract it (that is, find 8/9 of 8/9 of 4 Â D).2 2 Struve [1930, 168, 178] was perhaps the first to suggest such a half-circumference interpretation in regard to MMP 10. Neugebauer [1934, 124] remarked that such would indeed seem to be the case if Peet’s semicylinder interpretation of MMP 10 is accepted. 458 L. Cooper It can be seen that the final result of this process, when multiplied out, is equal to 256/ 81 Â D. For those who might wish a comparison in terms of p, one can rewrite 256/81 as 3 13/81, or as 3.1605. However, there is no written evidence that the scribes either thought in terms of, or worked things out in terms of, the specific concept we call p. It may be of benefit to reframe this discussion in terms of the supposition that if the scribes had desired to compute the circumference of a circle whose diameter was equal to 1 unit in length (D = 1), using both their unit fraction format and the algorithm proposed as being in evidence in MMP 10, they could have determined the circle’s circumference by calculating the eight-ninths part of the eight-ninths part of 4 Â 1, and thus have arrived at a unit-fraction answer of 3 + 1/9 + 1/27 + 1/81 — an answer which in a modern compound fraction format is equivalent to 256/81.3 If the circumference algorithm given here is essentially what the scribe was attempting to achieve in lines 5 through 11 of MMP 10, then it would seem that at least two significant questions must follow.
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