"Information Flows in Ancient Egyptian Arithmetic: a New Methodology" (2013)

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Information Flows in Ancient Egyptian Arithmetic: a New Methodology

Luca Miatello

Abstract

In ancient Egyptian mathematics, the algorithmic structure of the problem texts is characterized by the

presence of two levels of calculation: the main algorithmic level, constituted by a series of operations

executed step by step, and a second, “nested”, level of calculation, in which the individual operations

of multiplication and division are executed in a scheme organized on two-columns. While for the main

algorithmic level a methodology of mathematical rewriting that parallels the procedures of the ancient

Egyptians is available, no completely effective methodology has been proposed for the “nested” level

of calculation, which has been frequently read by means of anachronistic equations. The present article

aims to fill this gap. The information flows develop along two directions: vertical and

horizontal. Horizontal relations are in general implicit relations generated by operations of doubling,

halving, etc., but, in some cases, both horizontal and vertical flows are involved in the procedures.

This constitutes a sharp contrast with our modern, “one-way”, mentality.

Introduction: the two levels of calculation in Egyptian mathematics

The Egyptian mathematics is characterized by the presence of two levels of calculation. An

example is Problem 50 of the Rhind mathematical papyrus (BM EA 10057–8, dated to the

Second Intermediate Period), transcribed and translated in Table 1.1 As highlighted in Table

2, the first algorithmic level, which can be represented as a symbolic algorithm by means of

the method proposed by Ritter [1989] and applied to the main sources by Imhausen [2003],

consists of three instructions of calculation, executed step by step; two of them are

multiplications and are executed in a second level of calculation. The second, “nested”, level

of calculation is characterized by a different structure: the values in the operations of

1 Hieratic text: http://www.britishmuseum.org/collectionimages/AN00766/AN00766114_001_l.jpg; Robins and

Shute 1987, pl. 16. Hieroglyphic transcription, transliteration and German translation in Imhausen 2003: 248–50.

An Egyptian fraction, the reciprocal of an integer, is rendered in transcriptions as an integer with an overbar: 2

for one-half, 3 for one-third, etc., while the special sign for two-thirds is written 3 . The publication of the

Rhind mathematical papyrus, accompanied by hieratic transcription, was first proposed in Chase, Bull, Manning

and Archibald 1927/29.

http://www.britishmuseum.org/collectionimages/AN00766/AN00766114_001_l.jpg

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multiplication and division are written on two columns.2 In the case of Table 1, the values 1

and 8 are successively doubled.

tp n jr.t AH.t dbn n xt 9 Method of calculating a circular area of 9 xt ptj rx.t=f m AH.t Which is its measurement in area?

xbj.xr=k 9 =f m 1 Then you subtract its 9 as 1.

DA.t m 8 The remainder is 8.

jrj.xr=k wAH-tp m 8 zp 8 Then you multiply 8 by 8.

xpr.xr=f m 64 Then 64 results.

rx.t=f pw m AH.t sTA.t 64 Its measurement in area is 64 sTA.t

jr.t mj xpr Calculation as it develops,

1 9 1 9

9 =f 1 its 9 1

xb.t xnt=f DA.t 8 Subtraction from it, rest 8.

1 8 1 8

2 16 2 16

4 32 4 32

8 64 8 64

rx.t=f m AH.t sTA.t 64 Its measurement in area: 64 sTA.t

Table 1. Transliteration and translation of P. Rhind, No. 50.

Level 1 Level 2

9

↓

9 · 9 = 1 1 9

9 1

↓ ↓

9 – 1 = 8 9 – 1 = 8

↓ ↓

8 · 8 = 64 1 8

2 16

4 32

8 64

Table 2. The two algorithmic levels in P. Rhind, No. 50.

2 On the presence of two levels see also Ritter [2000, 124–25]. In the Moscow mathematical papyrus,

arithmetical operations are rarely executed by means of the two-columns scheme. The second level of

calculation is completely missing in Babylonian problem texts. Cf. Ritter 2004, 186.

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In the analysis of this level of calculation, the identification of the information flows is

frequently problematic: the relation between values is implicit, and part of the computational

process is occasionally left hidden. Purpose of this paper is to propose a new methodology to

represent the process of calculation arranged on two columns. Ritter [2000] proposed to add

wording (doubling, halving, etc.) to each line of the two columns, but this method is not

completely effective, especially in the case of complicated or peculiar procedures of

calculation. Anachronistic modern equations have been frequently used in mathematical

rewritings, with the consequent loss of relevant features of the original sources. In some

cases, the step-by-step algorithmic method can be used to highlight stages of calculation

(calculation of the lead term, "completion", etc.) [Miatello, 2007], but the two-columns

structure is not shown. Most of the recto and several exercises of the Rhind papyrus deal with

problems of “second level”, evidently considered of fundamental importance in the education

of an Egyptian scribe. Before entering into the details of the analysis, I will now discuss

briefly preliminary questions about the Egyptian system of numbers and its main features.

The Egyptian system of numbers

There is evidence, from hieroglyphic inscriptions in tags of bone and ivory from royal

tombs at Abydos and Naqada,3 that already in predynastic time, at the end of the fourth

millennium B.C., the Egyptians had developed a decimal system of numbers without

positional notation. Excluding administrative texts like the Abusir papyri, though, there is

scanty written evidence for the Old Kingdom mathematical practice, which, instead, must

have been considerable, on account of the huge human and material resources employed in

construction works. The earlier extant mathematical texts, written in hieratic on papyri,

leather, wood, are dated to the Middle Kingdom. The system of fractions displayed in such

texts is also very ancient: archaic hieroglyphic notations of one-half, two-thirds and three-

fourth, from the early dynastic annals, appear on the Palermo stone [Ritter 1992: 27–28]. The sign

for three-fourth later practically disappeared, substituted by the sum of 2 and 4 . An Egyptian

fraction, defined as the reciprocal of an integer, is inscribed by overwriting an integer with the

mouth sign in hieroglyphic, with a dot in hieratic. Exceptions are the special signs for 3 (

3 Dreyer 1998, 137–45; Quibell 1904, pl. 43.

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, ), 2 ( , ), 4 ( , ), and in hieratic also for 3 ( ).4 A multiple of fraction is

expressed as a sum of fractions in descending order of magnitude. For instance, the

multiplication of 5 by 6 , or, equivalently, the division of 5 by 6, is written as the sum of 2

and 3 , i.e., 2 3 . Doublings of fractions whose reciprocal is an odd number are executed and

listed in the opening section of the Rhind papyrus as divisions of 2 by odd numbers from 3 to

101 (“2/n table”).5 The results of such operations appear also in part in the Lahun fragment

UC 32159 [Imhausen and Ritter 2004, 92–93]. The multiplication of two fractions is in general

performed as the multiplication of two integers, e.g.: 2 · 5 = 10 , 3 · 6 = 9 .6 When 3 is

multiplied by a fraction whose reciprocal is not a multiple of 2, however, the operation makes

use of a particular technique, which is explained, with the example of 3 multiplied by 5 , in P.

Rhind, No. 61b: “Then you multiply by its 2 and its 6” (jr.x<r>=k zp 2=f zp 6=f ).7 This

means that 5, the reciprocal of 5 , has to be multiplied by 2 and 6. Implicitly, the instruction

suggests to take the reciprocals of the resulting integers, 10 and 30. In practice, 3 is

transformed into the sum 2 6 , which, multiplied by 5 , yields 10 30 .8 Several multiplications

of 3 by a fraction are collected in a table (P. Rhind, No. 61).

Within the capacity measuring system, fractions of the corn measure HqAt are expressed by the

signs 2 ’, 4 ’, 8 ’, 16 ’, 32’, 64 ’.9 There was no sign, for instance, for 3 HqAt, or for measures

lower than 64 HqAt. To fill the gaps, as we will see, the Egyptians used a combination of

fractions of HqAt and a measure called rA, “part”, written in the ordinary numerical system. The

HqAt contained 320 rAw, therefore, for example, 64 ’ corresponded to 5 rAw.

The two-columns scheme in operations of multiplication and division: preliminary

concepts

As previously mentioned, multiplications and divisions are typically performed in the

problem texts by means of a two-columns scheme, characterized by intermediate operations

4 On hieratic numerical signs: Möller 1909, 59–63, 65.

5 For a list of the results in the “2/n table”, which appear in the first line of each division: Imhausen 2007, 20.

6 The direct operation 3 · 6 = 9 is found in P. Rhind, No. 18. Cf. Peet 1923a, pl. G, 57.

7 Hieratic text: Robins and Shute 1987, pl. 18. Hieroglyphic transcription, transliteration, translation: Imhausen

2003, 269.

8 Imhausen 2003: 87. This is also the result of the division of 2 by 15 in the “2/n table”.

9 For the hieratic signs in the Rhind papyrus, see Imhausen 2003, 58. See also Möller 1909, 67.

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executed sequentially, step by step. Such scheme can be introduced with the example of 3

multiplied by 4, in P. Rhind, No. 26 (see Table 3). The unit and the multiplicand 3 are

doubled until, in the first column, the value 4, i.e., the second multiplicand, is obtained; the

corresponding value in the second column is the result 12. In the next two columns is also

shown the same multiplication, with the multiplicands exchanged. In this case, an additive

process is implied: the multiplicand 3 is reached as the sum of 1 and 2, and the result is the

sum of 4 and 8. The scheme is generalized in modern form in the last two columns: the

multiplicand b and the result R are obtained by means of appropriate multipliers k of the value

of initialization 1 and the multiplicand a.

3 · 4 4 · 3 a · b

1 3 \ 1 4 1 a

2 6 \ 2 8 1 · k1 a · k1

\ 4 12 3 12 … …

1 · k1 … · k n a · k1 … · k n

Σ = b Σ = R

Table 3. Examples of Egyptian multiplications, with a modern generalization.

The scheme of division can be described with the example of 12 divided by 3, in the first two

columns (see Table 4).

12 : 3 12 : 4 a : b

1 3 \ 1 4 1 b

2 6 \ 2 8 1 · k1 b · k1

\ 4 12 3 12 … …

1 · k1 … · k n b · k1 … · k n

Σ = R Σ = a

Table 4. Examples of Egyptian divisions, with a modern generalization.

The unit and the divisor 3 are successively doubled until, in the second column, the value of

the dividend (= 12) is obtained; the corresponding value in the first column is the result, i.e.,

4. In the next two columns is the division of 12 by 4: the divisor 4 is doubled and the value 12

is obtained as the sum of 4 and 8; the result is the sum of 1 and 2 in the first column, i.e., 3.

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The schemes of division and multiplication are analogous, but the result R and the target

value, namely the value to be reached by means of the multipliers k, are on opposite columns.

Typical values chosen by the Egyptians for the multipliers are 2, 10, 3 , 2 , 10 , but other

values could be used.10

Note that the values in the proposed examples of multiplication and

division are identical: the operations can be distinguished only by knowing the nature of the

given values (multiplicand, dividend or divisor), and/or the result. Furthermore, the relations

between values in the two-columns scheme are not clearly expressed: the multipliers are not

written, their presence is implicit. In order to highlight the information flows and fully

appreciate the complexity of the scheme, I will propose a method to represent the operations

of multiplication and division. The various examples of operations that will be proposed

contain the following mathematical terminology and grammatical formulae: 11

jr.xr=k N r gm.t M, “Then you divide M by N”.

wAH-tp m N r gm.t M, “Divide M by N”.

njs M xnt N, “Divide M by N”.

jr.t N zp M r gm.t K, “Multiply N by M to find K”.

xr N m wAH Hr=f, “Then N is added to it”.

tp n skm.t, “Method of completion”.

tp n sjtj, “Method of verification”.

gmi N, “Find N”.

DA.t, “Remainder”.

dmD, “Sum”, “total”.

dmD pA aHa N, “Total, this quantity: N”.

xpr.xr N, “Then N results”.

sSm.t, “Scheme”, “working out”.

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See, for instance, the multiplication in Table 1, where 9 is used as multiplier. Peet’s assertion that “The

Egyptian only multiplied directly by two figures, 2 and 10” [Peet 1923b, 92] is an excessive simplification,

although the purpose of the two-columns scheme is undoubtedly to perform the operations by means of typical

multipliers.

11 For a table of grammatical formulae expressing mathematical operations in the Lahun fragments: Imhausen

and Ritter 2004, 73.

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Information flows in operations of multiplication: the parallel vertical flows

The term sSm.t is used in Middle Egyptian mathematical texts to indicate a “scheme”, or

“working out”, with reference to both the two-columns scheme (e.g., P. Rhind, No. 41–44,

“2/n table”) and a sequence of operations executed step by step (P. Rhind, No. 65–66). In the

two-columns scheme, an operation of multiplication implies the execution of a sequence of

operations, guided by strategies. A simple example is the multiplication of 25 by 20 in P.

Rhind, No. 46 (Table 5).12

25 · 20 Rewriting

1 25 1 (∙25) 25

10 250

20 500 10 “ 250

\ 20 “ \ 500

Table 5. Multiplication in P. Rhind, No. 46, with information flows.

A method of rewriting of the two-columns scheme, here proposed on the right of the original

arrangement of numbers, highlights the information flows, indicated by an arrow next to a

multiplier. The flows develop along the vertical direction. There are also implicit horizontal

relations of multiplication, indicated in round brackets, or by the sign “. There is no doubt that

these horizontal relations were perceived by the Egyptians: as we will see, they were

commonly highlighted in the “2/n table”, and sometimes “horizontal multiplications” were

executed. In the scheme of Table 5, parallel vertical flows are generated by decupling and

then doubling both values in the first row. The first step is the initialization: the number 25 is

written next to the number 1 in the first row. The second step consists in multiplying both

numbers by 10, result 10 and 250. Finally, their parallel doubling gives the target value 20 on

one column, and the final result 500 on the other.13

Two typical multipliers are used: the

12

Hieratic text: Robins and Shute 1987, pl. 15. Hieroglyphic transcription, transliteration, translation: Imhausen

2003, 242–43.

13 The number 1 as value of initialization is usually written as a dot. In Ritter [2000, 127], the three steps are

written: “initialization”, “decupling”, “doubling”.

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multiplier 10 is the half of the target value 20, and the parallel multiplication by 2 serves as

completion. The final result in operations of multiplication is always on the second column,

while it is always on the first one in operations of division. An example of additive process is

the multiplication of 12 by 12 in the Rhind papyrus, Problem 32 (Table 6).14

The parallel

vertical flows, consisting in successive doublings of the numbers 1 and 12 in the first line, end

when the number 8 is obtained on the left column, since the sum of the intermediate results 4

and 8 equals the target value. The sum (dmD) of the corresponding intermediate results 48 and

96 in the second column gives the final result of 144. In this representation of the information

flows, the numbers adding up to the target value and to the final result are both marked with a

backslash, in order to facilitate the reading, while the Egyptians marked the entire line with a

single backslash.

12 · 12 Rewriting

1 12 1 (∙12) 12

2 24

\ 4 48 2 “ 24

\ 8 96

dmD 144 \ 4 “ \ 48

\ 8 “ \ 96

(12) 144

Table 6. Multiplication in P. Rhind, No. 32, with information flows.

In the previous examples, the parallel vertical flows are continuous: the successive

application of the multipliers to the values in the first row yields the final result. In other

cases, more than one set of parallel vertical flows are identifiable. For instance, in P. Rhind,

No. 53/54, the multiplication of 7 sTA.t (measure of area) by 2 4 is executed by identifying two

target values: 2 and 4 (Table 7).15

The first target value is reached by doubling the

initialization values, 1 and 7. Then, in order to find the second target value, a new set of

parallel vertical flows are developed, by successively halving the initialization values. The

14

Robins and Shute 1987, pl. 11; Imhausen 2003, 216–17.

15 Robins and Shute 1987, pl. 16; Imhausen 2003, 254.

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indirect execution of horizontal operations, by simple parallel operations of doubling, halving,

etc., can be convenient for both integers and fractions.

7 sTA.t · 2 4 Rewriting

1 7 sTA.t 1 (∙7) 7

\ 2 14 sTA.t

2 3 2 sTA.t \ 2 “ \ 14

\ 4 1 2 4 sTA.t •

dmD 15 2 4 sTA.t

2 “ 3 2

\ 4 “ \ 1 2 4

(2 4 ) 15 2 4

Table 7. Multiplication in P. Rhind, No. 53/54, with information flows.

A parallel level of calculation: the auxiliary numbers in an operation of addition

The example of the multiplication of 4 28 by 1 2 4 , in P. Rhind, No. 7,16

allows to

introduce the important concept of “auxiliary numbers” (Table 8). In the first row, the first

multiplicand is written next to the number 1, which is one of the key values. Since the other

two target values are 2 and 4 , parallel vertical flows of successive halvings are executed. The

result is the sum of six fractions on the second column. While the sum and subtraction of

integers is always obtained directly, the sum and subtraction of fractions, when they are not

straightforward, require the conversion of the fractions into auxiliary numbers. An appropriate

multiplier is chosen, for instance the inverse of one of the fractions, in this case 28 , which

means that the auxiliary 1 is assigned to 28 . The sum 4 28 is multiplied by 28, result 7 + 1;

the sum 8 56 is transformed into 3 2 + 2 ; the sum 16 112 into 1 2 4 + 4 . Each auxiliary is

written in the text in red ink under the corresponding original value. The sum of the

auxiliaries, omitted in the papyrus, is 14, and the transformation into the true sum is given by

16

http://www.britishmuseum.org/collectionimages/AN00766/AN00766119_001_l.jpg; Robins and Shute 1987,

pl. 9; Peet 1923a, pl. G, 54.


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the division of 14 by 28, i.e. 2 . There are two parallel levels in this process: the usual scheme

of multiplication, with the calculation of the results corresponding to the target values, and the

transformation of the intermediate results into auxiliaries. Note that vertical information flows

are used in this case even in the calculation of the auxiliaries: the transformation into

auxiliaries is evidently executed indirectly for 8 56 , and for 16 112. Only the transformation

of 4 28 into 7 + 1 is executed by direct multiplication by 28. For instance, the result of the

multiplication of 8 by 28 is obtained simply by halving the auxiliary 7. The hypothesis

formulated by Robins and Shute [1987: 19] that it was obtained directly by using the two-

columns scheme is unnecessarily involved. The title, tp n skm.t, “method of completion”,

signifies that the auxiliaries allow to “complete” (skm) a number of fractions to a value to be

determined. In this case, the sum of fractions adds up to 2 . As we will see, the technique of

the auxiliary numbers is used also to subtract a sum of fractions from a given value.

4 28 · 1 2 4 Rewriting

tp n skm.t

1 4 28 \ 1 (∙ 4 28 ) \ 4 28 7 + 1

7 1

2 8 56 \ 2 “ \ 8 56 “ 3 2 + 2

3 2 2

4 16 112 \ 4 “ \ 16 112 “ 1 2 4 + 4

1 2 4 4

dmD 2 (1 2 4 ) 2 (14)

Table 8. Multiplication and completion in P. Rhind, No. 7, with information flows.

Information flows in operations of division

As previously mentioned, a division is performed as a form of multiplication. For instance,

the division of 15 by 5, in P. Rhind, No. 26,17

is executed by operating on 5 in order to find 15

(Table 9).18

The value next to 1 in the first row is always the divisor, and the target value is

the dividend, which is found typically by using the multipliers 2, 10, 3 , 2 and 10 . In the

17

Robins and Shute, 1987, pl. 10; Imhausen 2003, 208.

18 The literal translation of the instruction in the text is “calculate with 5 to find 15”.

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operations of division the result is always on the left, and the “key lines”, corresponding to the

values whose sum gives the target value, are marked with a backslash.

As in the case of the multiplication, more than one set of parallel vertical flows can be

implied in a division. In P. Rhind, No. 25,19

16 is divided by 3 (Table 10). The target value 16

is approximated by the sum of 3 and 12, after the execution of successive doublings.

15 : 5 Rewriting

wAH-tp m 5 r gm.t 15

\ 1 5 \ 1 (∙5) \ 5

\ 2 10

xpr.xr 3 \ 2 “ \ 10

3 (15)

Table 9. Division in P. Rhind, No. 26, with information flows.

16 : 3 Rewriting

\ 1 3 \ 1 (∙3) \ 3

2 6

\ 4 12 2 “ 6

3 2

\ 3 1 \ 4 “ \ 12

•

3 “ 2

\ 3 “ \ 1

5 3 (16)


At this point, since 15 is 1 short of the target value, one would expect to find the values 3 and

1 in the fourth line of the operation. Instead, the value 1 is reached by halving 3 of 3: a

19

Robins and Shute 1987, pl. 10; Imhausen 2003, 207.

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second set of parallel vertical flows is developed, by using the typical multipliers 3 and 2 .

Curiously, the Egyptians preferred not to calculate directly the third of a quantity, which was

obtained convolutedly, by halving the typical multiplier 3 .

The “inversion of flow” and the technique of “inversion-multiplication-inversion”

When the target value cannot be reached by means of parallel vertical flows, a particular

technique is used. An example of its application is the division of 320 (rAw) by 11 (Table 11),

from the Cairo wooden tablets.20

320 : 11 Rewriting

1 11 \ 1 (∙11) \ 11

10 110

20 220 10 “ 110

2 22

4 44 \ 20 “ \ 220

8 88 •

11 1

2 “ 22

4 “ 44

\ 8 “ \ 88 (Σ = 319)

\ 11 \ 1

2911 (320)

Table 11. Division in the Cairo wooden tablets, with information flows.

In order to obtain the target value 320, the divisor 11 is multiplied by the typical multipliers

10 and 2, to give the intermediate result 220; then a second set of parallel vertical flows,

20

The tablets (CG 25367–8, Middle Kingdom) show how to calculate the third, seventh, tenth, eleventh,

thirteenth, of a whole hekat: Daressy 1901, 95–96. See Vymazalova 2002, 27–42; Peet 1923b.

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consisting in successive doublings, yields the value 88. The sum of 220, 88 and 11, is 319,

which is 1 short of the target value, thus the final step consists in inverting 11. The result of

the division is hence 2911 (= 1 + 20 + 8 + 11). As usual, parallel vertical flows allow to

execute indirectly the horizontal operations.21

The rewriting of the procedure on the right of

Table 11 highlights the technique of “inversion of (horizontal) flow”, which allows to find the

last of the values adding up to the result: an arrow from right to left indicates the inversion.

The Rhind papyrus shows several operations of division in which the technique of

inversion is applied, in particular in the “2/n table”. In the case of 2 divided by 5,22

the target

value (= 2) is approximated by means of parallel vertical flows, consisting in applying the

typical multipliers 3 and 2 to the unit and the given divisor (Table 12).

2 : 5 Rewriting

5 3 1 3 15 3 sSmt 1 5 1 (∙5) 5

3 3 3

\ 3 1 3 3 “ 3 3

\ 15 3

\ 3 “ \ 1 3

\ 15 \ 3

3 15 (2)

Table 12. Division in P. Rhind, recto, with information flows.

This process yields the first term of the decomposition, 3 , and, in the second column, the

value 1 3 . The remainder, which completes to 2, is 3 , and the inversion of flow implies the

multiplication of 3 by 5 , result 15. The final result 3 15, marked with backslashes, is written

in the text also in red ink, in the first row. Such a row highlights the “horizontal flows” of the

“key lines”, marked by backslashes: 3 multiplied by 5 gives 1 3 , and 15 multiplied by 5 is 3 .

21

The “key lines” are not marked with backslashes in the text. Despite the loss of legibility, this omission is

frequent.

22 http://www.britishmuseum.org/collectionimages/AN00238/AN00238262_001_l.jpg; Robins and Shute 1987,

pl. 2; Peet 1923a, pl. A; Vogel 1929, 116.


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In the case of 2 divided by 7 (Table 13),23

the approximation to the dividend is performed

by means of parallel vertical flows, successively halving the values 1 and 7 in the first line,

until 1 2 4 is reached. The inversion of flow is performed as follows: the reciprocal of the

remainder is 4, which is multiplied by the divisor 7, by successively doubling 1 and 7; the

result is then inverted. In the rewriting of the information flows, on the right side, this

“inversion-multiplication-inversion” technique is represented by the notation : the right

sign ^ indicates the inversion of 4 ; the left sign ^ denotes the inversion of the result of 4

multiplied by 7.

2 : 7 Rewriting

1 7 4 1 2 4 28 4 1 (∙7) 7

2 3 2 1 7

\ 4 1 2 4 2 14 2 “ 3 2

\ 4 28 4 \ 4 28

\ 4 “ \ 1 2 4

\ 28 \ 4

4 28 (2)


A further example of this technique is the division of 10 by 3 10 in P. Rhind, No. 30 (Table

14).24

Parallel vertical flows are developed, by successive doublings, until the sum 9 3 10 10

15 30 is obtained. The remainder is 30 .25

To perform the inversion of flow, the text suggests

the “multiplication of 30 by 23, to find 3 10 ”. This presupposes the “inversion-

multiplication-inversion” technique: multiplication of 30 by 3 10 , and inversion of the result

23. The final result of the operation of division is hence 13 23. Note that the results for twice

23


pl. 1; Peet 1923a, pl. A; Vogel 1929, 116.

24 Robins and Shute 1987, pl. 10; Imhausen 2003, 212.

25 The multiplication of 10 10 15 30 by 30 gives 9, therefore the addition of 30 makes 10 times 30 , or 3 .


15

5 and twice 15 can be obtained from the “2/n table”. The notation 2↓* indicates this

derivation.

10 : 3 10 Rewriting

jr.xr=k 3 10 r gm.t 10

\ 1 3 10 \ 1 (∙3 10 ) \ 3 10

2 1 3 5

\ 4 315 2 “ 1 3 5

\ 8 610 30

dmD 13 (DAt) 30 \ 4 “ \ 315

jr.t 30 zp 23 r gm.t 3 10

dmD pA aHa Dd sw 13 23 \ 8 “ \ 610

30 (Σ = 9 3 10 10 15 30)

\ 23 \ 30

13 23 (10)


The inversion of flow as intermediate technique and its indirect execution

In the previous examples, the inversion of flow is the final technique of the computational

procedure, but, in some cases, it is an intermediate technique. For example, in the division of

10 by 1 2 4 , executed in P. Rhind, No. 34 (Table 15),26

the successive doubling of 1 2 4 yields

the value 7, which, added to 1 2 4 , gives 8 2 4 ; the target value 10 is hence reached by adding

1 4 . The technique of inversion of flow is used with reference to the value 4 , and it is

plausible to assume that the “inversion-multiplication-inversion” is here implicitly applied:

the reciprocal of 4 , i.e. 4, is multiplied by 1 2 4 , and the result 7 is then inverted. Ritter called

“inversion” this case, represented as N · a = M → M · a = N [Ritter 2000, 127–28], which can

be ascribed, more in general, to the technique of inversion of flow. Obviously, we do not

know whether the scribe here writes first 4 , and then 7 , or vice versa, but the introduction of

26

Robins and Shute 1987, pl. 12; Imhausen 2003, 221.

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4 , as value necessary to reach the target value, is procedurally antecedent to the calculation of

7 . A further important technique is used in this division to perform the inversion of flow with

reference to the value 1, which completes to 10. The division of 1 by 1 2 4 would have

required a further operation by means of the two-columns scheme. Instead, knowing from the

“2/n table” that twice 7 is 4 28 , it is sufficient to double successively the values 7 and 4 to

obtain 2 14 and 1 on the two columns. The inversion of flow is thus conveniently executed

indirectly, by means of successive doublings.

10 : 1 2 4 Rewriting

\ 1 1 2 4 \ 1 (∙1 2 4 ) \ 1 2 4

2 3 2

\ 4 7 2 “ 3 2

\ 7 4

4 28 2 \ 4 “ \ 7

\ 2 14 1

DmD pA aHa 5 2 7 14 \ 7 \ 4

4 28 2

\ 2 14 \ 1

5 2 7 14 (10)


The indirect execution of the inversion of flow, by using the “2/n table”, allows to

complete the result of a division when the rule “inversion-multiplication-inversion” does not

yield a fraction. A further example of this technique is the division of 100 by 7 2 4 8 , in P.

Rhind, No. 70 (Table 16).27

27

Robins and Shute 1987, pl. 19; Imhausen 2003, 288–90.

17

100 : 7 2 4 8 Rewriting

jr.xr=k wAH-tp m 7 2 4 8 r gm.t 100

1 7 2 4 8 1 (∙7 2 4 8 ) 7 2 4 8

2 15 2 4

\ 4 31 2 2 “ 15 2 4

\ 8 63

\ 3 5 4 dmD 99 2 4 DAt 4 \ 4 “ \ 31 2

63 8 qAb tjt r 4

\ 42 126 4 \ 8 “ \ 63

•

\ 3 “ \ 5 4 Σ = 99 2 4

63 8

\ 42 126 \ 4

12 3 42 126 (100)


Two sets of parallel vertical flows, one by successive doublings, the other by employing the

multiplier 3 , allow to determine the additive quantity 99 2 4 , and then the rest 4 .28

The

multiplication of the reciprocal 4 by 7 2 4 8 yields 31 2 , which is not an integer and cannot be

directly inverted. To solve this problem, the value 8 is introduced, in order to “double the

figure to 4 ” (qAb tjt r 4 ). The multiplication of 8 by 7 2 4 8 gives 63, and the number 63 is

written on the left column, as result of the inversion of the horizontal flow. Finally, parallel

vertical flows, by means of the multiplier 2, yield the rest 4 on the second column and, by

deriving the result of twice 63 from the “2/n table”, the value 42 126 on the first column. The

result of the division is given by the sum of the key values, highlighted by backslashes.29

28 The multiplication of 7 2 4 8 by 3 is presumably executed as division of 15 2 4 by 3.

29 The same technique to perform indirectly the inversion of flow is found in the Cairo wooden tablets for the

divisions of 320 (rAw) by 13 and 7. Cf. Vymazalova 2002, 32–34, 37–38.

18

An operation of subtraction: ‘completion’ by means of auxiliary numbers

As in the case of the addition of fractions, the subtraction from a given value can be

conveniently performed by means of auxiliary numbers. The technique is described in P.

Rhind, Nos. 21–23. Problem 21, for instance, requests to complete (skm) the quantity 3 15 to

1 (Table 17).30

Dd n=k skm m-a

3 15 m 1

10 1

dmD 11 DA.t m 4

wAH-tp m 15 r gm.t 4

1 15

10 1 2

\ 5 3

\ 15 1

dmD 4

xr 5 15 m wAH Hr=f tp n sjtj

xr km 3 5 15 15 r 1

10 3 1 1

Table 17. Transcription of P. Rhind, No. 21.

The procedure is composed of four stages (Table 18): 1) the transformation of the given

sum of fractions into auxiliary numbers, by means of a common multiplier; 2) the execution

of a subtraction; 3) the execution of a division; 4) the verification of the completion by means

of auxiliary numbers. In the first stage, the quantity 3 15 is transformed into the sum of 10 (=

3 · 15) and 1 (= 15 · 15), i.e. 11. The second stage consists in subtracting 11 from 15, result

4; then the division of 4 by 15 yields the quantity 5 15 , to be added to 3 15 (xr 5 15 m wAH

Hr=f). Finally, each of the four fractions is multiplied by 15, in order to verify that their sum

“is complete to 1” (km 3 5 15 15 r 1). Correctly, the sum of their auxiliaries is 15.

Auxiliaries are therefore used in two different contexts: in the procedure of “completion”, and

in the method of verification (tp n sjtj).

30


pls. 9–10; Imhausen 2003, 203.


19

3 10

15 1

4 = 15 – 11

4 : 15 = 5 15

3 15 + 5 15 = 1

3 10

5 3

15 1

15 1

Table 18. Procedural steps in P. Rhind, No. 21.

This technique is presumably implied in several divisions of 2 by odd numbers, when a

direct completion to the target value is not straightforward, for instance in the division of 2 by

41 (Table 19).31

2 : 41

njS 2 xnt 41 24 1 3 24 246 6 328 8

sSmt 3 27 3 1 41

3 13 3 \ 2 82

6 6 3 6 \ 4 164

12 3 3 12 dmD \ 6 246 6

\ 24 1 3 24 \ 8 328 8

DA.t 6 8

Table 19. Division in P. Rhind, recto.

Parallel vertical flows, by means of the multiplier 3 and successive halvings, allow to

approximate the target value 2 to 1 3 24 (Table 20).

31


pl. 4; Peet 1923a, pl. B; Vogel 1929, 122.


20

(1) (∙41) (41)

3 “ 27 3

3 “ 13 3

6 “ 6 3 6

12 “ 3 3 12

\ 24 “ \ 1 3 24 [ 3 16

24 1

\ 246 \ 6 [7 : 24] 7 = 24 – 17]

\ 328 \ 8

24 246 328 (2)

Table 20. Information flows in the division of 2 by 41.

The lead term is therefore 24 . The completion of 3 24 to 1, which is not shown in the text,

can be performed by means of auxiliary numbers,32

which lead to the division of 7 by 24,

result 6 8 .33

Then the “inversion-multiplication-inversion” technique gives the second and

third term of the decomposition, i.e., 246 and 328 . As always in the divisions of 2 by odd

number in the Rhind papyrus, there is an initial line, in which “horizontal flows” of the “key

lines” are highlighted: the numbers whose sum is the result of the division are written in red

ink, and next to each of this number is the result of its multiplication by 41.

32

The general use of predetermined identities collected in tables to perform the “completion” has been suggested

by some scholars (e.g., Knorr 1982, 136–37), but the Mathematical Leather Roll contains a very limited number

of identities. Also, the skm.t-problems of the Rhind papyrus provide a general method. It is not excluded, though,

that tables of sum of fractions adding up to the unit were also used.

33 7 (: 24) is the sum of 4 (: 24) and 3 (: 24). An alternative result of this division is 4 24 .

21

An example of particularly complex division

Some operations of division in the Rhind papyrus are distinguishable by their complexity.

For example, in P. Rhind, No. 31,34

various techniques, including the auxiliary numbers, are

applied in order to divide 33 by 1 3 2 7 (Table 21).

33 : 1 3 2 7

1 1 3 2 7

\ 2 4 3 4 28

\ 4 9 6 18sic

\ 8 18 3 7

2 2 3 4 14

\ 4 4 6 8 28 dmD 32 2 DAt 2

\ 97 42

\ 56 679 776 21

\ 194 84

\ 388 168

7 8 14 28 28

6 5 4 3 1 2 1 2

17 4

3 2 4 2 21 dmD 33

1 42

3 28

2 21

7 6 dmD 99sic

Table 21. Division in P. Rhind, No. 31.

It is possible to distinguish three stages of calculations. In the first stage, described in Table

22, the target value 33 is approximated, as always, by means of appropriate multipliers. Three

34


pl. 11; Imhausen 2003: 214–15.


22

successive doublings of the divisor 1 3 2 7 lead to the sum 32 3 .35

It is easy to ascertain that

the technique of inversion cannot be applied successfully by using the remainder 3 , hence a

closer approximation to 33 is to be attempted. The addition of half the given divisor to 32 3

yields a value higher than 33, hence a further halving is to be performed. The sum of the key

values on the second column at this point is 32 2 7 8 14 28 28 , and the rather equivocal claim

“sum 32 2 , rest 2 ” means that the total sum is broken into two parts: 32 2 and 7 8 14 28 28 ,

which is to be completed to 2 .

1 (∙1 3 2 7 ) 1 3 2 7

\ 2 “ \ 4 3 4 28

\ 4 “ \ 9 6 14

\ 8 “ \ 18 3 7 (Σ = 31 3 4 6 7 14 28 = 32 3 )

•

2 “ 2 3 4 14

\ 4 “ \ 4 6 8 28 Σ = 32 2 (+ 7 8 14 28 28 ); 33 – 32 2 = 2

Table 22. Information flows in the division of 33 by 1 3 2 7 : first stage.

The second stage of the process is therefore the completion of this sum of fractions to 2 (see

Table 23).36

35

This intermediate sum, which is not shown in the papyrus, can be calculated by using the identity 7 14 28 = 4

, which appears in the Mathematical Leather Roll. On this table of 26 identities between a sum of fractions and a

single fraction, found with the Rhind papyrus: Glanville 1927; Imhausen 2007, 21.

36 The auxiliaries calculated by Neugebauer [1931: 337] give a “rest” of 28 that is not shown in the papyrus.

23

7 8 14 28 28 6 + 5 4 + 3 + 1 2 + 1 2

Σ = 17 4

2 21

21 – 17 4 = 3 2 4

Table 23. Information flows in the division of 33 by 1 3 2 7 : second stage.

Significantly, the value 42, which is a multiple of 6 and 7, is chosen as multiplier: as we will

see, its multiplication by 1 3 2 7 yields the integer number 97 and, therefore, allows to

perform the “inversion of flow”. The sum of fractions is transformed into the auxiliary

quantity 17 4 , and the completion to 2 requires the division of 3 2 4 (= 21 – 17 4 ) by 42.

The third stage is implicit in the text, but, in accordance to the previously highlighted

process of completion, consists in determining the unit fractions of the completion and the

“inversion of flow”, as represented in Table 24. Noticeably, the multiplication of 3 2 4 by 42

can be executed by means of the two-columns scheme, by partitioning the target value 3 2 4

into 1 + 2 + 2 + 4 . The first value of completion is hence 42 , and the inversion of flow gives

97 on the opposite column. As shown in the text, the multiplication of 42 by 1 3 2 7 yields

the integer number 97 (erroneously written 99). The other key values are obtained by means

of parallel vertical flows of doubling and halving, deriving the result for twice 97 from the

“2/n table”.

... ...

\ 97 \ 42 ( 1)

(∙2)

\ 56 679 776 \ 21 ( 2)

\ 194 “ \ 84 ( 2 )

(∙ 2 )

\ 388 \ 168 ( 4 )

14 4 56 97 194

388 679 776 (33)

Table 24. Information flows in the division of 33 by 1 3 2 7 : third stage.

24

This and other divisions in the Rhind papyrus are characterized by a high level of

sophistication. Much easier are the divisions in the “2/n table”, which occupies about half of

the recto. We have already examined some of the techniques involved in the division of 2 by

5, 7, and 41. Further techniques and procedures of calculation are described, although not

always in detail, in this section of the papyrus.

Further procedures and techniques in the “2/n table” of the Rhind papyrus

In the division of 2 by 101, which appears on the small fragments of the Rhind papyrus in

the Brooklyn Museum,37

the lead term is the reciprocal of the given divisor, and the

predetermined identity 2 3 6 = 1, which is calculated in Problem 16 of the papyrus, is used as

completion, by means of three inversions of the horizontal flow (“inversion-multiplication-

inversion” technique). Restored text and information flows are illustrated in Table 25.

2 : 101 Rewriting

[... 101 1] 202 2 033[ 3 606 6 ] \ 101 (∙101) \ 1

sSmt [\ 101 1]

\ 2 [202] 2 \ 202 \ 2

\ 3 303 3 \ 303 \ 3

\ 6 606 6 \ 606 \ 6

101 202 303 606 (2)

Table 25. Division in P. Brooklyn (fragments of P. Rhind, recto), with information flows.

A further technique, which allows to find directly the first term of the decomposition, is

applied in the “2/n table” with reference to divisors that are multiples of 3. In the case of 2

divided by 27, for instance (Table 26),38

the value 3 indicates that its multiplication by 27

gives 18, whose reciprocal is the first term of the decomposition.

37

Brooklyn Museum 37.1784E: Robins and Shute 1987, pl. 8; Peet 1923a, pl. D; Vogel 1929, 128.




25

2 : 27 Rewriting

1 27 18 1 2 54 2 1 (∙27) 27

\ 3 18 1 2 3 \ 18 \ 1 2

\ 2 54 2

\ 54 \ 2

18 54 (2)


This technique can be defined “ 3 rule”: the reciprocal of the lead term is 3 of the divisor. The

value 3 as multiplier that allows to find directly the first term of the decomposition appears in

the “2/n table” for all the divisors multiples of 3, with the exception of 9, 15 and 93, but even

in these cases the reciprocal of the first term is always 3 of the divisor.

For the other divisors of 2 that are composite numbers (25, 49, 55, 65, 77, 85, 95), a

procedure of derivation from prime numbers was presumably used, although no direct textual

evidence is available. In the division of 2 by 65, for instance (Table 27), considering that the

divisor 65 is equal to 5 multiplied by 13, it is sufficient to use the already available result of 2

divided by 5, i.e. 3 15, to find the first term of the decomposition, simply dividing 3 by 13.39

2 : 65 Rewriting

njs 2 xnt 65 39 1 3 195 3 [(2 : 5) : 13]

sSmt gmi 39 1 3

\ 3 195 3 [ 3 ] [: 13] \ 39 \ 1 3

[15] “ \ 195 \ 3

39 195 (2)

Table 27. Division in P. Rhind, recto; hypothetical information flows.

39

On the probable derivation of composites from prime numbers, see, for example: van der Waerden 1980, 265–

66.

26

The lead term 39 , multiplied by 65, gives 1 3 , and then completion and inversion of flow lead

to the second terms of the decomposition, i.e. 195. According to this hypothesis, for

composite numbers but the multiples of 3, the lead term was derived from the already

available results of the division of 2 by prime number. The completion can be executed

directly, or by derivation.

An alternative hypothesis has been recently formulated: various lead terms that multiplied by

65 yield values less than 2 would have been introduced directly (e.g., 33 , 34 , 35 , etc.).

Supposing that two-terms decompositions were preferred, after experiments we would find

39 195. This hypothesis, formulated by Abdulaziz as a “general method” for all the divisors

of 2 [Abdulaziz, 2008], is laborious, since the completion to 1 is to be calculated in any viable

case; it is impossible to know a priori if the remainder is composed of one, two, or three

fractions, and the search process does not end when the first two-terms result is found.40

Furthermore, the three-terms decomposition 60 380 570 was chosen to represent 2 divided by

95, despite the availability of the two-terms decomposition 57 285.

The divisions of 2 by 35 and 91 are peculiar: the solutions are not derivable from a prime

number divisor, and the multiplication of the second term of the decomposition by the divisor

of 2 yields a sum of two fractions, and not a single fraction. For the divisor 35, the papyrus

shows the terms 30 and 42 (dot omitted), which, multiplied by 35, yield 1 6 and 3 6

respectively (Table 28).41

2 : 35

35 30 1 6 42 3 6

6 7 5

\ 30 1 6

\ 42 3 6


40

For instance, in the case of the divisor 77, a possible solution is 42 462, but the papyrus shows 44 308 .




27

In the first row, the numbers 7 and 5 are written in black under the two terms of the

decomposition, while the number 6 is written in red under the divisor 35. The calculation is

then proposed by means of the two-columns scheme. Two main procedures have been

suggested to explain this division and, as we will see, the division of 2 by 91.

The first hypothesis, schematized in Table 29, implies the application of the skm.t-method.42

The numbers under the result would be auxiliaries in a verification, by using the multiplier

210. The choice of a multiplier yielding integer numbers appears, however, unusual.43

Also, it

is unclear how the inversion of flow would have been performed. The inversion of 5/6,

followed by the multiplication (or division) of the non-Egyptian fraction 6/5 by 35

[Neugebauer 1926, 36, note 1; Robins and Shute 1987, 27; Abdulaziz 2008, 10], is questionable. For

instance, in P. Rhind, No. 67, the multiplication of 3 by 3 gives 6 18 (= 2/9), which, divided

into 1 by means of the two-columns scheme, yields 4 2 . Significantly, the result was not

obtained simply as the reciprocal of 2/9. This case seems to suggest that the scribe would

have attempted the inversion of flow by dividing 1 by 3 6 , result 15 , which, multiplied by 35

gives 42.

2 : 35, first hypothesis:

\ 30 \ 1 6 [ 6 1

\ 42 \ 3 6 [5 : 6] 5 = 6 – 1]

30 42 (2)

30 7

42 5

35 6

Table 29. Hypothetical information flows.

As we will see, though, the “inversion-multiplication-inversion” technique is fruitless when

applied to the analogous case of 2 divided by 91. One should also consider that auxiliary 42

See, for example, Robins and Shute 1987, 26–27; Abdulaziz 2008, 10.

43 In the Ostracon 153 from the tomb of Senmut, the value 7 , its double (= 6 14 21), and quadruple (= 2 14 ) are

accompanied by their transformation into auxiliaries, written in red ink, by means of the multiplier 21: Hayes

1942, pl. 29; Gillings 1972, 87.

28

numbers do not appear elsewhere in the “2/n table”, and the fact that they are used in a

peculiar case makes the assumption that they served only as verification of the result unlikely.

Their presence is probably more significant, in accordance with the second hypothesis,

represented in Table 30.44

Since the divisor 35 equals 7 multiplied by 5, the division of 2 by

35 can be transformed, by means of the multiplier 6, into 12 divided by 210. The partition of

12 into 7 and 5 allows to find directly the two terms of the decomposition: the divisions of 7

and 5 by 210 yield 30 and 42 respectively. The multiplication of the two terms by 35 gives 1

6 and 3 6 , whose sum is 2. A similar algorithm is described in the Demotic papyrus BM EA

10520, from the Roman period.45

2 : 35, second hypothesis:

[12] : (35 · 6)

7 [: 210] \ 30 \ 1 6

5 [: 210] \ 42 \ 3 6

30 42 (2)

Table 30. Hypothetical information flows.

The division of 2 by 91 from the recto of the Rhind papyrus is shown in Table 31.46

2 : 91

91 70 1 5 10 130 3 30

gmi \ 70 1 5 10

gmi 130 3 30


44

This interpretation was first proposed in Vogel 1929, 157–160. See also van der Waerden 1980, 268.

45 Parker 1972, 65–66. It is interesting to note that the procedure to divide 2 by 35 is described step-by-step, as in

the main algorithmic level.

46 Robins and Shute 1987, pl. 7; Peet 1923a, pl. D.; Vogel 1929, 127.

29

Considering the first hypothesis (Table 32), the completion to 1, by means of auxiliary

numbers (multiplier 10), gives 3 30 , but the attempt to invert the horizontal flow, with the

division of 1 by 3 30 , etc., gives no obvious result. On the other hand, the division of the non-

Egyptian fraction 7/10 by 91, to yield 130, is questionable. In this case, auxiliary numbers are

not found in the text, but the word gmi, “find”, is peculiarly written twice, before both terms

of the decomposition. Such occurrence can be interpreted as the scribe’s suggestion to

calculate the terms in succession, in accordance with the second hypothesis (Table 33): the

division of 2 by 91 is transformed into the division of 20 by 910, by means of the multiplier

10, which is the half of the sum of 7 and 13. Since 7 and 13 are divisors of 91, their division

by 910 yields the two terms of the decomposition.

2 : 91, first hypothesis:

\ 70 \ 1 5 10 [ 5 2

10 1

\ 130 ? \ 3 30 [7 : 10] 7 = 10 – 3]

70 130 (2)

Table 32. Hypothetical scheme of division.

2 : 91, second hypothesis:

[20 : (91 · 10)]

[13 : 910] \ 70 \ 1 5 10

[7 : 910] \ 130 \ 3 30

70 130 (2)

Table 33. Hypothetical scheme of division.

Table 34 shows synthetically how the lead term and the terms after the first were

calculated in the “2/n table”, for all the divisors of 2. In almost all cases, the techniques are

directly described, or the calculations can be ascribed to techniques illustrated in other

sections of the papyrus. Note that this paper discusses the procedures of calculation, without

30

investigating for which reason a particular decomposition was preferred to possible others.47

The values of the first term referred to the divisors 43, 59, 97 – namely 42 , 36 , 56

respectively –, indicate that experiments were not always conducted by using typical

multipliers. This supports Peet’s assertion that the collection of values was the result of an

empirical accumulation, although not necessarily protracted over a long period [Peet 1923a, 36].

The use of several procedures for the same problem type is typical of the Egyptian mathematics.

Examples are given by the algorithmic variety (of first level) in the “aha problems” [Imhausen

2003, 35–53], and by the two procedures in the Cairo wooden tablets.

Divisors of 2 First term Terms after the first

3 3 —

5, 7, 11 Found by successive halving of

3 , 2 , 3 Found by direct completion to 2

13, 17, 19, 23, 29, 37, 41 Found by successive halving of

2 , 3 (?), 3 , 3 , 3 , 3 , 3

Found by completion to 1,

probably involving a division

31, 43, 47, 53, 59, 61, 67,

71, 73, 79, 83, 89, 97

20 , 42 , 30 , 30 , 36 , 40 , 40

40 , 60 , 60 , 60 , 60 , 56

Found by completion to 1,

probably involving a division

101 101 Found by predetermined identity

2 3 6 = 1

9 Found by successive halving of

3 (inverse of 3 of the divisor)

Found by direct completion

1 2 + 2 = 2

15, 21, 27, 33, 39, 45, 51, 57,

63, 69, 75, 81, 87, 93, 99 Inverse of 3 of the divisor Found by direct completion

1 2 + 2 = 2

25, 49, 55, 65, 77, 85, 95 Derived from prime divisors (?):

5, 7, 11, 5, 7, 5, 19 Found by direct completion to 2

35, 91 Found by multiplier and partition

technique (?)

Found by multiplier and partition

technique (?)

Table 34. Techniques of calculation in the “2/n table” of the Rhind papyrus.

The two procedures in the Cairo wooden tablets

Two procedures are used in the Cairo wooden tablets to transform non-canonical fractions

of HqAt into sums of HqAt fractions and rAw.48

The first procedure is composed of two stages:

division of 320 rAw by 7 (or by 10, 11, 13); transformation of the result into a sum of HqAt

47

On canons of selection of the values in the “2/n table”, see Abdulaziz 2008, 1–18; Miatello 2007, 327–47.

48 Cf. Vymazalova 2002; Peet 1923b. Excepting the division by 10, the operations appear repeatedly on the

tablets: twice for the division by 3, three times for the division by 13, four times for the division by 7 and 11.

31

fractions and rAw. For example, the division of 320 by 11 yields 2911 (see Table 11); then 29

11 is transformed into 16 ’ 64 ’ + 411 rAw.

Significantly, the procedure to divide the HqAt by 3 is peculiar (see Table 35):

multiplication of 3 by 5 rAw; multiplication of the result by 64, by means of successive

doublings and transformations into sums of HqAt fractions and rAw.

1 3

2 3

4 1 3

64 ’ 1 3

32’ 3 3

16 ’ 64 ’ + 1 3

8 ’ 32’ + 3 3

4 ’ 16 ’ 64 ’ + 1 3

2 ’ 8 ’ 32’ + 3 3

\ 1’ 4 ’16 ’ 64 ’ + 1 3

\ 2’ 2 ’ 8 ’ 32’ + 3 3

Table 35. Division of the HqAt by 3, in the Cairo wooden tablets.

In the first stage (Table 36), the multiplication of 3 by 5 (rAw) is executed by means of

successive doublings. The result, 1 3 (rAw), is the sum of 3 and 1 3 .

In the second stage (Table 37), 1 3 rAw is multiplied by 64. Such multiplication is executed

by employing 64 ’, the equivalent of 5 rAw, as value of initialization, instead of the usual

value 1. The transformations into HqAt fractions and rAw occur during the process of

successive doubling: twice 3 3 rAw is 6 3 rAw, which is written as the sum of 64 ’ and 1 3 rAw.

Successive doublings, accompanied by metrological transformations, continue until a whole

HqAt is obtained on the first column, to which corresponds, on the second column, the

requested result. A final doubling serves as verification that the sum of the result of 3 HqAt

and its double is equal to one HqAt.

It is interesting to note that there is no apparent advantage in employing this alternative

procedure: the application of the procedure used in all the other cases would have implied the

simple division of 320 (rAw) by 3, followed by metrological transformations.

32

3 · 5

\ 1 (·3 ) \ 3

2 “ 3

\ 4 “ \ 1 3

(5) 1 3

Table 36. Information flows in the division of the HqAt by 3, first stage.

1 3 (rAw) · 64

64 ’ (·3 ) 1 3

32’ “ 3 3

16 ’ “ 64 ’ + 1 3

8 ’ “ 32’ + 3 3

4 ’ “ 16 ’ 64 ’ + 1 3

2 ’ “ 8 ’ 32’ + 3 3

\ 1’ (·3 ) \ 4 ’16 ’ 64 ’ + 1 3

\ 2’ “ \ 2 ’8 ’ 32’ + 3 3

Table 37. Information flows in the division of the HqAt by 3, second stage.

Conclusions

I have proposed a new methodology to highlight the information flows in the Egyptian

two-columns scheme of calculation, which develops along two directions, vertical and

33

horizontal. Horizontal relations are in general implicit relations generated by operations of doubling,

halving, etc. These “horizontal flows” are highlighted in an initial line for each division in the “2/n

table” of the Rhind papyrus. In some cases, horizontal multiplications are involved in the procedures

of calculation, and, in operations of division, the horizontal flow could be appropriately inverted.

The Egyptians used both horizontal and vertical directions, in contrast with our modern, “one-way”,

mentality. A parallel level of calculation is represented by the auxiliary numbers, which are

used in various stages of the computational process: calculation of the sum of fractions;

subtraction of a fraction or a sum of fractions by a given value; verification (sjtj) of a result. A

division is performed by using the two-columns scheme, but also, when the dividend is a

fraction, by means of the “inversion-multiplication-inversion” technique. Such a technique is

used to invert the information flow in the two-columns scheme. An indirect method of

inversion of flow consists in utilizing the results of the “2/n table” in processes of successive

doubling. The four arithmetical operations are characterized by reciprocal influences:

multiplications and divisions imply an additive process; divisions often require subtractions;

additions and subtractions of fractions make use of multiplications and divisions, etc. In some

cases, the reconstruction of the information flows in the two-columns scheme is

straightforward; in others, when the procedure of calculation is partially left hidden,

techniques of calculation described in other sections of the papyrus, as in the case of skm.t-

methods, provide a plausible explanation. There are however processes of uncertain

interpretation, in which the comparison of alternative hypotheses can be useful. The proposed

method of representation of the operations of calculation allows to avoid the loss of the

characteristics of the original sources caused by the use of modern equations, or other

abridgements of the procedures, without renouncing to propose an interpretation.

Typical of the Egyptian mathematics is the use of several procedures for the same problem

type. In particular, multiple procedures are used in the Rhind papyrus to divide 2 by odd

numbers, and in the Cairo wooden tablets to execute metrological transformations. In the use

of various procedures and techniques one may individuate not only a flexible response to the

complexity in specific situations, but also a wider philosophical attitude, in accordance with

the Egyptian view of a universe ordered by multiple gods, in which various laws coexisted,

even in conflict one another.

The new methodology proposed in this paper highlights complexity and peculiarity of the

Egyptian computational processes, constituted by horizontal and vertical information flows,

inversions of flow, levels of calculation, techniques and strategies. Such elaborate processes

34

have little in common with modern arithmetic, but allowed to perform difficult operations,

providing basic techniques to be employed for all relevant needs of an Egyptian scribe, in

administration or construction.

Considering that even the algorithmic method proposed by Ritter for the main level of

calculation, after more than two decades has not yet been adopted by all researchers, I doubt

that the present method of rewriting of arithmetic operations will receive unanimous

consensus, but I hope that the present article can at least contribute to the knowledge of the

Egyptian procedures of calculation.

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"Information Flows in Ancient Egyptian Arithmetic: a New Methodology" (2013)

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