Rhind Mathematical Papyrus, From Wikipedia,

The Rhind Mathematical Papyrus (RMP) has been housed at the British Museum since 1864. It is named after Alexander Henry Rhind a modern era owner, as well as after an ancient scribe Ahmes. Rhind a Scottish antiquarian purchased the papyrus in 1858 in Luxor, Egypt. It was apparently found during illegal excavations near the Ramesseum. The British Museum, where the papyrus is now kept, acquired the RMP in 1864, along with the Egyptian Mathematical Leather Roll (EMLR). Both documents were gifted to the British after the untimely death of Henry Rhind. There are small fragments of the RMP held by the Brooklyn Museum in New York. It is one of two well-known Mathematical Papyri. The second is the Moscow Mathematical Papyrus (MMP). The RMP is larger than the MMP, while the latter is older than the former. [1]

The RMP dates to the Second Intermediate Period of Egypt and is the best example of Egyptian mathematics. The RMP may have been partly copied by its scribe Ahmes (i.e., Ahmose; Ahmes is an older transcription favoured by historians of mathematics) from a now-lost text from the reign of king Amenemhat III (12th dynasty), an assumption that may not be true. Written in hieratic script the RMP is 33 cm tall and over 5 meters long, and began to be transliterated, and mathematically translated in the late 19th century and early 20th century. The document dates to Year 33 of the Hyksos king Apophis and contains a later date, Year 11 on its verso likely from his successor, Khamudi.[2]

In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving - Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries…all secrets.

Contents: 1 Mathematical problems, 2. Mathematical knowledge, 3. Influence of the RMP.

1. Mathematical problems

The papyrus was written on both sides. It began with a RMP 2/n table, followed by 87 problems. Taking one third of the manuscript, the RMP 2/n table expressed 2 divided by the odd numbers from 5 to 101 in terms of unit fraction series, written in increasingly larger denominators. Unit fractions were not repeated. Ahmes conversion method was parsed three fragmented ways by 20th century and earlier researchers. One way was identified by F. Hultsch in 1895, and confirmed by E.M. Bruins in 1945. Today the aliquot part fragments is called the H-B method. A second way was red auxiliary numbers noted as least common multipliers (by Gillings and others). Two of the ways are implicitly used in the Egyptian Mathematical Leather Roll, selecting single and double reference numbers. A third way, considered in 2001, factored 2/95 = (1/5)(2/19) and converted by the Hultsch-Bruins Method with red auxiliary numbers. Improved translations of the 2/n table and its ancient scribal meanings have been proposed after 2008.

Ahmes rational number conversion method used red auxiliary numbers as defined by RMP 36, a central fact that is needed to be considered when proposing scribal meanings of the 2/n table. Ahmes converted n/p and 2/n vulgar fractions to concise, not not always optimal, Egyptian fraction series by selecting a LCM m. The LCM m scaled n/p to mn/mp and 2/n to 2n/nm. The best divisors of denominators that summed to numerators mn and 2n were written in red. The red auxiliary numbers allowed concise unit fraction series to be recorded. In 2008, it was clear that the red auxiliary number method allowed Ahmes to convert 2/95 by selecting 12 as a LCM 12 such that:

(2/95)(12/12) = 24/(12)(95) = (19 + 3 + 2)/1140 = 1/60 + 1/380 + 1/570

Confirmation of the 2008 analysis is also obtained by 2009 data from RMP 36. The 2008 and 2009 analyses showed that Ahmes frequently omitted all three of the intermediate steps. The exception is RMP 3 when one and one-half of the intermediate steps were omitted,

RMP 41, RMP 42 and RMP 43 applied algebra to solve three geometry formula problems, one for the area of a circle and two for volumes of circular granaries.

Confusions for over 100 years surrounded Ahmes’ calculations causing 20th century math historians to improperly personalize versions of Ahmes’ ’ab initio’ method. For example, Gillings in 1972 only reported the LCM multiplier step. Several 20th century historians’ suggested other ’ab initio’ methods. Scholars came close to Ahmes’ calculation method, pointing out one, two and even aspects of the third step, but not all three steps, in a clear manner, until 2008.

Appying Ahmes’ three-step red reference number method, the RMP’s 84 problems began with six division by 10 problems, the subject of the Reisner Papyrus, RMP 1-6 selected optimized red reference numbers to calculate quotient and exact remainder answers. Confusingly, Ahmes’ shorthand notes only outlined duplation proofs. Ahmes showed that n/10 divisions were returned to unity by multiplying answers by its divisor (a point mentioned by Robins-Shute in 1987, but missed by the majority of math historians). There were 15 problems that dealt with addition, and 18 algebra problems. There were 15 algebra problems of the same type. The algebra problems asked the reader to find x and a fraction of x such that their sum equals an integer.

RMP 33 solved

x +(2/3 + 1/2+ 1/7)x = 37

x + (55/42)x = 37

(97/42)x = 37

x = 1554/97

as modern arithmetic operations require today. Ahmes further considered

x = 16 + 2/97

as a quotient 16 and remainder 2/97, converting 2/97 to an Egyptian fraction series as the 2/n table reports by selecting 56 as a red LCM number, mentally, or elsewhere, computing:

(2/97)(56/56) = 112/(97)(56) = (97 + 8 + 7)/(5432) = 1/97 + 1/679 + 1/77

proved by duplation notes that: 16 + 1/97 + 1/679 + 1/776

was the arithmetically correct answer. The duplation notes began with the answer, a procedure that reported the contents of previous 32 RMP problems, meaning that ’ab initio’ calculations were not presented by Ahmes.

Each of other 14 algebra problems implicitly converted vulgar fraction answers in the same manner. Vulgar fraction answers were converted to optimized Egyptian fraction series by selecting red reference numbers, with each answer discussed in an explicit duplation proof. Duplation multiplication proofs varied in completeness, Ahmes omitted important arithmetical facts. For example, in RMP 23 45 was suggested as the LCM, rather than 360, Ahmes actual LCM.

In RMP 41 and 42 Ahmes modified an area (A) of a circle formula by replacing radius (R) with diameter (D/2) and pi by 256/81 and added volume (V) by multiplying by height (H) such that

Sqrt(A) =(8/9)(D) cubits squared (algebraic geometry formula 1.0)

reported in the Moscow Mathematical Papyrus 300 years earlier, and

V = (3/2)[(8/9)(D)(8/9)(D)] cubits cubed (algebraic geometry formula 1.1)

RMP 43 scaled algebraic geometry formula 1.1 by 3/2 considering

(3/2)Sqrt(A) = (3/2)(8/9)(D) = (4/3)D

and

V = [(2/3)H(4/3)(D)(4/3)(D)]khar (algebraic geometry formula 1.2)

a formula reported in the Kahun Papyrus 250 years earlier.

Two arithmetical progressions (A.P.) were solved in RMP 40 and RMP 64. A common method followed had been defined in the Kahun Papyrus. RMP 64 problem solved the A.P. sharing 10 hekats of barley, between 10 men, by a difference of 1/8th of a hekat finding 1 7/16 as the largest term.

The second A.P. problem was RMP 40. The problem divided 100 loaves of bread between five men such that the smallest two shares were 1/7 of the largest three shares’ sum (87 1/2). Ahmes found the shares for each man, which he did without finding the difference (9 1/6) or the largest term (38 1/3). All five shares 38 1/3, 29 1/6, 20, 10 2/3 1/6, and 1 1/3) were calculated by first finding the five terms from a proportional A.P. that summed to 60. The median and the smallest term, x1, were used to find the differential and each term. Ahmes then multiplied each term by 1 2/3 to obtain the sum to 100 A.P. terms. In reproducing the problem in modern algebra, Ahmes also found the sum of the first two terms by solving

.

In RMP 81, 29 examples of Akhmim Wooden Tablet partitioned a hekat unity (64/64) by rational numbers 1/64 ¡ n ¡ 64. In addition 15 RMP problems were similar to Moscow Mathematical Papyrus problems. Finally, 23 RMP weights and measures problems often discussed the hekat, with three recreational diversion problems following, the last being RMP 84, the famous multiple of 7 riddle, passed down to the Medieval era as As I was going to St. Ives from other sources.

The Rhind Mathematical Papyrus also contains geometry related problems:[3]

”If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?”

The solution to the problem is given as the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face.[3]

2. Mathematical knowledge

Upon closer inspection, modern-day mathematical analysis of Ahmes’ problem-solving strategies reveal a basic awareness of composite and prime numbers;[4] arithmetic, geometric and harmonic means;[4] a simplistic understanding of the Sieve of Eratosthenes[4], and perfect numbers.[4][5]

The papyrus also demonstrates knowledge of solving first order linear equations[5] and summing arithmetic and geometric series.[5]

The papyrus calculates Pi as (8/9) squared, times 4, or 3.1605 (a margin of error of less than 1 per cent). Two viable theories offering some insight into a possible motivation for such an accurate derivation have been proposed

1. African crafts demonstrating snake curves and sets of equidistant concentric circles, and 2. Board game resembling Mancala, found in the Mortuary Temple of Seti I (both boardgames utilize small and large circles; see Mancala).

Other problems in the Rhind papyrus demonstrate knowledge of arithmetic progressions, algebra and geometry.

The papyrus also demonstrates knowledge of weights and measures, business, and recreational diversions.

3. Influence of the RMP

The Egyptian use of arithmetic proportions are briefly discussed by Gillings. In particular the use of the Remen, which has two values, is reflected in the foot which has two values, (the second being the nibw or ell which is two feet), and the cubit which has two values. Doubling is also seen in the subdivisions such as fingers and palms. Since doubling seems to have been the basis of most of the unit fraction calculations, which it was not (least common multiples were) up to and including the calculations of circles with dimensions given in khet (see Ancient Egyptian units of measurement), looking at how the remen and seked were used provided many insights to Greek and Roman geometers and architects. The actual and proposed readings/decodings of the RMP and Kahun 2/n tables is required to be fairly interjected.

In the Rhind Papyrus we encounter the remen which is defined as the proportion of the diagonal of a rectangle to its sides when its other sides are whole units. Yet, a singular arithmetic proportion formula reported in the RMP and Kahun Papyrus offer an additional example beyond the remen’s diagonal of a square, with its sides a cubit. We also find problems using the seked or unit rise to run proportion. Typical of the Classical orders of the Greeks and Romans, it was built upon the canon of proportions derived from the inscription grids of the Egyptians.

This document is one of the main sources of our knowledge of Egyptian mathematics.