Moscow Mathematical Papyrus

Legend

• All the scripts (including the diagrams) are flipped to be read left to right.
• Column numbers are according to [Struve 1930].

<red

red>

tp-ḥsb

Rubrum. Written in red.

m[j ḫpr]

j ḫpr restored from being partially or almost entirely damaged.

<1/>10

The scribe omitted the glyph.

ḫpr-ḫr

The scribe wrote an extra glyph.

|3|

Line number.

Problem 1

1

<red

red>

2

3

Problem 2

1

<red

red>

2

3

Problem 3

1

<red

red>

2

3

4

5

Problem 4

1

<red

(triangle)

red>

2

(triangle)

3

4

5

6

(triangle partially damaged; not likely a right triangle)

•

•

＼

Problem 5

1

<red

red>

2

3

4

5

6

7

8

9

10

(1/2)

11

(1/2)

12

13

14

Problem 6

1

<red

red>

^[1]

2

^[1]

3

4

5

6

＼

＼

|1| tp n jr.t ꜥ.t[1]
|2| mj ḏd n=k ꜥ.t[1] nt sṯꜣt 1/2 1/4 n ꜣw n wsḫ
|3| jrj.ḫr=k jr=k 1/2 1/4 r gm.t wꜥ(1) ḫpr.ḫr [m] 1 1/3
|4| [jrj 12] pn ntj m sṯꜣt 1 1/3 ḫpr.[ḫr] 16
|5| jrj.ḫr=k jr=k qnb.t=f ḫpr.ḫr 4 n ꜣw 1/2 1/4 m 3 n |6| wsḫ jr.t mj ḫpr
    4
 +----+
 | 12 | 3
 +----+
\1  4
\2  8

Notes

1. ꜥ.t is reading of [Gunn & Peet 1929]. [Imhausen 2002] doesn't agree that

   

   is present in line 1, but agrees with them in line 2. [Struve 1930] reads it as
   pr.

Problem 7

Column 9

1. <red

   (triangle)

   red>

2. (triangle)

Problem 8

1

<red

red>

2

3

^[1]

4

^[2]

5

6

7

^[3]

8

(1/2)

9

10

11

Notes

1. For

   

   , [Struve 1930] reads

   

   .
2. For

   

   , [Peet 1931] suggests thta it is the scribe's typographic error of

   

   .
3. For
   , [Peet 1931][Peet 1931] reads
   .

Problem 9

1

<red

M33A

red>

2

M33A

3

4

5

6

7

8

M33A

9

M33A

M33A

10

M33A

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

M33A

M33A

27

28

29

30

Problem 10

1

<red

red>

2

3

4

5

6

^[1]

7

8

9

10

11

12

13

14

|1| tp n jr.t nb.t
|2| mj ḏd n=k nb.t m tp rꜣ |3| r 4 1/2 m ꜥḏ
ḥꜣ |4| dj=k rḫ=j ꜣḥ.t=s
jrj.ḫr=k |5| jrj=k 1/9 n 9 ḥr-ntt jr nb.t
|6| 1/2 pw n j[...][1]
ḫpr.ḫr 1
|7| jrj.ḫr=k jrj=k ḏꜣ.t m 8
|8| jrj.ḫr=k jrj=k 1/9 n 8
|9| ḫpr.ḫr 2/3 1/6 1/18
jrj.ḫr=k |10| jrj=k ḏꜣ.t n.t pꜣ 8 r sꜣ |11| pꜣ 2/3 1/6 1/18
ḫpr.[ḫr] 7 1/9
|12| jrj.ḫr=k jrj=k 7 1/9 zp 4 1/2
|13| ḫpr.ḫr 32
mk ꜣḥ.t=s pw
|14| gmj=k nfr

Notes

1. For the damaged word, [Struve 1930] reads
   jnr. [Peet 1931b] denies the possibility and suggests
   jpt.

Problem 11

1

<red

red>

2

3

4

^[1]

5

^[1]

6

7

8

9

10

^[1]

11

Problem 12

1

<red

M33A

red>

2

M33A

3

4

5

6

7

8

9

10

11

Problem 13

1

<red

M33A

red>

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Problem 14

1

<red

⏢

red>

2

⏢

3

4

5

6

7

8

9

10

11

12

|1| tp n jr.t ⏢
|2| mj ḏd n=k ⏢ n 6 n sttj
|3| [r] 4 ḥr ẖr.y r 2 ḥr ḥr.w
|4| [jrj].ḫr=k jrj=k 4 pn m znn
ḫpr.ḫr 16
|5| jrj.ḫr=k qꜣb=k 4
ḫpr.ḫr 8
|6| jrj.ḫr=k jrj=k 2 pn m znn
ḫpr.ḫr 4
|7| jrj.ḫr=k dmḏ=k pꜣ 16 |8| ḥnꜥ pꜣ 8 ḥnꜥ pꜣ 4
|9| ḫpr.ḫr 28
jrj.ḫr=k jrj=k |10| 1/3 n 6
ḫpr.ḫr 2
jrj.ḫr=k |11| jrj=k 28 zp 2
ḫpr.ḫr 56
|12| mk nj-sw 56
gmj=k nfr

Problem 15

Column 30

1. <red

   M33A

   red>

2. M33A

Problem 16

1

<red

red>

2

3

4

^[2]

5

6

(1/2)

7

8

9

(1/4) (1/8)

10

11

Notes

1. is per [Imhausen 2002], reading

   

   as a ligature according to [Möller 1909]. [Struve 1930] reads
   , and [Peet 1931] reads
   .

2. 
   

   is according to Imhausen. [Struve 1930] reads

   
   

   , and [Peet 1931] reads

   
   
   

   .

Problem 17

1

(triangle)

2

(triangle)

3

4

5

6

7

8

9

•

＼

Problem 18

Column 35

|1| tp n jr.t rꜣ n [...] 5 [...] m sttj
|2| mj ḏd n=k rꜣ n [...] 5 [...] jrj m sttj
|3| ḥꜣ dj=k rḫ=j sttj=f
jrj.ḫ<r>=k jrj=k |4| pꜣ 5 [...]
ḫpr.ḫr 53 m pꜣ 5
|5| ḫpr.ḫr 10 m pꜣ rꜣ
jrj.ḫr=k jrj=k 51 [...][1] zp

Notes

1. [Struve 1930] reads it as
   m 80, [Miatello 2013] reads it as
   ntf mnḫ.t.

Problem 19

Column 36

Problem 20

Column 37

5. M33A

   (1/4)

   (1/16)

   (1/64)

Problem 21

1

2

(1/8)

(1/16)

3

(1/8)

4

5

(1/16)

6

7

8

Problem 22

1

M33A

2

M33A

3

4

5

6

7

M33A

8

9

10

(1/2)

Problem 23

Column 42

Problem 24

1

M33A

2

M33A

3

4

5

6

7

8

9

10

11

Problem 25

Column 45

References

• Imhausen, A. (2002). Ägyptische Algorithmen: eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten (Vol. 65). Otto Harrassowitz Verlag.
• Miatello, Luca. "A Longstanding Enigma: Problem 18 of the Moscow Mathematical Papyrus", Journal of the American Research Center in Egypt, 2012.
• Miatello, Luca. "Problem 10 of the Moscow Mathematical Papyrus: Corrupted Part or Technicality?", Göttinger Miszellen, 2013.
• Miatello, Luca. "Inferring the Construction Process of Two Geometric Algorithms, Göttinger Miszellen, 2018.
• Peet, T. Eric. "A problem in Egyptian geometry." The Journal of Egyptian Archaeology, Vol. 17. pp. 100-106. 1931.
• Peet, T. Eric. "Notices on Recent Publications: Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau." The Journal of Egyptian Archaeology, Vol. 17. pp. 154-160. 1931.
• Struve, W. W. Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen. 1. Band. 1930.
• Moscow Papyrus Problem 10