List or table: T-S Ar.39.380
List or table T-S Ar.39.380What's in the PGP
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- 1 Translation
Description
Bifolio from a small booklet of mathematical (multiplication and division) tables in Coptic numerals, written in columns with ruled lines between groups of entries. Probably written by and/or for a merchant or fiscal official — someone who needed to multiply large sums and to divide by 12, 24, and 48, presumably for the purposes of converting dīnārs to dirhams (at the canonical rate of 1:12) and qīrāṭs (1:24). The first page contains part of a multiplication table in the "decadic" format: each multiplicand n is multiplied by k, 10k, 100k and 1000k (k = 1–9) in ruled groups of four, followed by a single line n × 10,000. Surviving entries run from 50 × 500 to 90 × 5000. The second page contains division tables for the divisors 12, 24 and 48 (divisors not written), applied to the dividends 1–9, 10–90, 100–900, 1000–9000 and 10,000, with quotients in unit fractions. Surviving entries run from 5 ÷ 12 to 9000 ÷ 48. There are also Arabic annotations. On p. 1 (fol. 1a), there is a note of several lines in the space where a fourth column of numerals would be expected, ending wa-l-salām and possibly also containing a name, including the patronym Ibn Azhar. On p. 3 (fol. 2a), there are two Arabic annotations. Following the ÷ 12 tables is the word al-qīraṭayn, "the two qīrāṭs," indicating that the column is dividing each number by 12 = two qīrāṭs. Following the ÷ 24 tables are the words ṣuwar al-qīrāṭ ("the figures for the qīrāṭ). The annotations indicate that one purpose of the division tables is to have a handy reference for converting quantities of dīnārs into qīrāṭs and that the table is, among other things, intended as a handy reference or teaching tool for commercial or fiscal purposes. Codicology: Coptic is read left to right and the order of pages are consistent with that, but this is not a continuous or inner bifolio. Pp. 2 and 3 face each other but are not continuous (p. 2 ends at 90 × 5000; p. 3 begins with the division tables), so at least one further bifolium was nested inside. P. 4 is written in a single column and is otherwise blank; p. 1 begins mid-way through the 50 table. The bifolium is therefore probably the outer bifolium of the final (or only surviving) quire of the booklet, whose text ended on p. 4. Goldstein and Pingree edited the fragment in summary as an appendix to their Geniza horoscopes, since it parallels a division-by-24 table in Coptic numerals in the margins of MS Strasbourg 4110. See also Sesiano, 59 no. 2 and 64 n. 6, who identifies the multiplication table with the dhāqāt, the second of two kinds of Coptic multiplication table described in an anonymous Arabic treatise on Coptic reckoning (Istanbul, Feyzullah Ef. 1365). This is the variant with 37 products per table, in which the final group of four is reduced to the product with 10,000. Sesiano places it in a tradition of Greek and Coptic tables that includes BL Or. 5707 and Greek papyri from the second to tenth centuries. Sesiano also cites this manuscript as an example of the Coptic myriad sign, a horizontal stroke with two dots beneath to indicate ten thousands (that unicode character does not render properly in PGP, so instead, my transcription uses a diaresis above, the Greek convention for myriads, e.g., 𐋡̈ = 10,000; 𐋮̈ = 500,000). Transcription conventions: ͵ before a numeral = thousands stroke (written to the lower left of the numeral). So, e.g., ͵ⲁ = 1,000 𐋡̈ (diaeresis) = myriad sign, e.g., 𐋡̈ = 10,000; 𐋮̈ = 500,000 𐅵 = ½ (S-shaped in this hand) 𐅷 = ⅔ (ω-shaped in this hand) ʹ to the upper right of a numeral = unit fractions. 𐋣ʹ = ⅓, , 𐋤ʹ = ¼, 𐋦ʹʹ = ⅙, 𐋨ʹ = ⅛, 𐋪𐋢ʹ = 1/12, 𐋫𐋤ʹ = 1/24 Horizontal lines in the manuscript are shown as — —. Notes on paleography: 𐋢 (2) and 𐋺 (800) are both ω-shaped (distinguished by context) 𐋶 (400) is a rounded u-shape 𐋮 (50) is U-shaped 𐋷 (500) is a crossed circle with an upright stroke 𐋯 (60) is a J-shaped hook 𐋸 (600) is 8-shaped 𐋹 (700) is cross-shaped 𐋩 (9) is a crossed circle with a horizontal bar 𐋲 (90) is a hook Marina Rustow (September 2026)
Editor: Rustow, Marina
Translator: Rustow, Marina
T-S Ar.39.380 1r
p. 2 (multiplication, 70, 80, and 90)
_Column 1_
(top lost: 70 × 1, 2, and 3)
1 [𐋰 𐋤 𐋴]𐋱
2 [𐋰 𐋭] ͵𐋢[𐋺]
3 𐋰 𐋶 𐋢̈ ͵𐋨
4 𐋰 ͵𐋤 𐋫̈ 𐋨̈
——
5 𐋰 𐋥 𐋵𐋮
6 𐋰 𐋮 ͵𐋣𐋷
7 [𐋰] 𐋷 𐋣̈ ͵𐋥
(bottom lost: 70 × 5000)
_Column 2_
(top lost: 70 × 6, 7, 8)
1 [𐋰 𐋩 𐋸]𐋬
2 [𐋰] 𐋲 ͵𐋦𐋵
3 𐋰 𐋻 𐋦̈ ͵𐋣
4 𐋰 ͵𐋩 𐋯̈ 𐋣̈
——
5 𐋰 𐋡̈ 𐋰̈
_Column 3_
(top lost: 80 × 1)
1 [𐋱 𐋢 𐋳]𐋯
2 [𐋱 𐋫] ͵𐋡𐋸
3 [𐋱 𐋴] 𐋡̈ ͵𐋦
4 [𐋱] ͵𐋢 𐋪̈ 𐋦̈
——
5 [𐋱 𐋣 𐋴]𐋭
6 [𐋱 𐋬] ͵𐋢𐋶
7 [𐋱] 𐋵 𐋢̈ ͵𐋤
8 [𐋱 ͵𐋣] 𐋫̈ 𐋤̈
——
9 𐋱 𐋤 𐋵[𐋫]
10 𐋱 𐋭 ͵𐋣𐋴
11 𐋱 𐋶 𐋣̈ ͵𐋢
12 𐋱 ͵𐋤 𐋬̈ 𐋢̈
——
13 𐋱 𐋥 𐋶
14 𐋱 𐋮 ͵𐋤
15 𐋱 𐋷 𐋤̈
16 [𐋱] ͵𐋥 𐋭̈
——
_Column 4_
1 [𐋱 𐋸] 𐋤̈ ͵𐋨
2 [𐋱 ͵𐋦] 𐋭̈ 𐋨̈
——
3 𐋱 𐋧 𐋷[𐋯]
4 𐋱 𐋰 ͵𐋥𐋸
5 𐋱 𐋹 𐋥̈ [͵𐋦]
6 𐋱 ͵𐋧 𐋮̈ [𐋦̈]
——
7 𐋱 𐋨 𐋸𐋭
8 𐋱 𐋱 ͵𐋦𐋶
9 𐋱 𐋺 𐋦̈ ͵𐋤
10 𐋱 ͵𐋨 𐋯̈ 𐋤̈
——
11 𐋱 𐋩 𐋹𐋫
12 𐋱 𐋲 ͵𐋧𐋴
13 𐋱 𐋻 𐋧̈ ͵𐋢
14 𐋱 ͵𐋩 𐋰̈ 𐋢̈
——
15 𐋱 𐋡̈ 𐋱 (should be 𐋱̈)
——
_Column 5_
1 𐋲 𐋳 [͵𐋩]
2 𐋲 ͵𐋡 𐋩̈
——
3 𐋲 𐋢 𐋳𐋱
4 𐋲 𐋫 ͵𐋡𐋺
5 𐋲 𐋴 𐋡̈ ͵𐋨
6 𐋲 ͵𐋢 𐋪̈ 𐋨̈
——
7 𐋲 𐋣 𐋴𐋰
8 𐋲 𐋬 ͵𐋢𐋹
9 𐋲 𐋵 𐋢̈ ͵𐋧
10 𐋲 ͵𐋣 𐋫̈ 𐋧̈
——
11 𐋲 𐋤 𐋵𐋯
12 𐋲 𐋭 ͵𐋣𐋸
13 𐋲 𐋶 𐋣̈ [͵𐋦]
14 𐋲 ͵𐋤 𐋬̈ 𐋦̈
——
15 𐋲 𐋥 𐋶𐋮
16 𐋲 𐋮 ͵𐋤𐋷
17 𐋲 𐋷 𐋤̈ ͵𐋥
18 [𐋲 ͵𐋥] 𐋭̈ [𐋥̈]
p. 3 (division by 12, 24, 48)
_Column 1 (÷ 12)_
(top lost: 1–4 ÷ 12)
1 𐋥 𐋣ʹ 𐋪𐋢ʹ
2 𐋦 𐅵
3 𐋧 𐅵 𐋪𐋢ʹ
4 [𐋨 𐅷]
5 𐋩 𐅵 𐋤ʹ
6 𐋪 𐅵 𐋣ʹ
7 𐋫 𐋡 𐅷
8 𐋬 𐋢 𐅵
9 𐋭 𐋣 𐋣ʹ
10 𐋮 𐋤 𐋦ʹ
11 [𐋯] 𐋥
12 𐋰 𐋥 𐅵 [𐋣ʹ]
13 𐋱 𐋦 𐅷
14 𐋲 𐋧 𐅵
15 𐋳 𐋨 𐋣ʹ
_Column 2 (÷ 12)_
(top lost: 200–800 ÷ 12)
1 𐋻 [𐋰𐋥]
2 ͵𐋡 𐋱[𐋣 𐋣ʹ]
3 ͵𐋢 [𐋳𐋯𐋦 𐅷]
4 ͵𐋣 𐋴𐋮
5 ͵𐋤 𐋵𐋬𐋣 [𐋣ʹ]
6 ͵𐋥 𐋶𐋪𐋦 [𐅷]
7 ͵𐋦 𐋷
8 ͵𐋧 𐋷𐋱𐋣 [𐋣ʹ]
9 ͵𐋨 𐋸𐋯𐋦 𐅷
10 ͵𐋩 𐋹𐋮
11 𐋡̈ 𐋺𐋬𐋣 [𐋣ʹ]
(Arabic note after column)
القيرطين
_Column 3 (÷ 24)_
(top lost: 1–20 ÷ 24)
1 𐋬 𐋡 [𐋤ʹ]
2 𐋭 𐋡 𐅷
3 𐋮 𐋢 𐋪[𐋢ʹ]
4 𐋯 𐋢 𐅵
5 𐋰 𐋢 𐅵 [𐋣ʹ 𐋪𐋢ʹ]
6 𐋱 𐋣 𐋣ʹ
7 𐋲 𐋣 𐅵 𐋤ʹ
8 𐋳 𐋤 𐋦ʹ
_Column 4 (÷ 24)_
(top lost: 200–2000 ÷ 24)
1 ͵𐋣 𐋳[𐋫𐋥]
2 ͵𐋤 𐋳𐋯𐋦 𐅷
3 ͵𐋥 𐋴𐋨 [𐋣ʹ] (?)
4 ͵𐋦 𐋴𐋮
5 ͵𐋧 𐋴𐋲𐋡 𐅷
6 ͵𐋨 𐋵𐋬𐋣 𐋣ʹ
7 ͵𐋩 𐋵𐋰𐋥
8 𐋡̈ 𐋶𐋪𐋦 𐅷
(Arabic note after column)
صور القيراط
_Column 5 (÷ 48)_
(top lost: 1–10 ÷ 48)
1 𐋫 [𐋤ʹ 𐋦ʹ]
2 𐋬 𐅵 𐋨ʹ
3 𐋭 [𐅵 𐋣ʹ]
4 𐋮 𐋡 [𐋫𐋤ʹ]
5 [𐋯] 𐋡 [𐋤ʹ]
6 𐋰 𐋡 [𐋣ʹ 𐋨ʹ]
7 𐋱 𐋡 𐅷
8 𐋲 𐋡 𐅵 [𐋣ʹ 𐋫𐋤ʹ]
(bottom lost: 100 ÷ 48)
p. 2 (multiplication, 70, 80, and 90)
Column 1_
(top lost: 70 × 1, 2, and 3)
1 [70 × 4 = 2]80
2 [70 × 40] = 2,[8]00
3 70 × 400 = 28,000
4 70 × 4,000 = 280,000
——
5 70 × 5 = 350
6 70 × 50 = 3,500
7 [70] × 500 = 35,000
(bottom lost: 70 × 5000)
_Column 2_
(top lost: 70 × 6, 7, 8)
1 [70 × 9 = 6]30
2 [70] × 90 = 6,300
3 70 × 900 = 63,000
4 70 × 9,000 = 630,000
——
5 70 × 10,000 = 700,000
_Column 3_
(top lost: 80 × 1)
1 [80 × 2 = 1]60
2 [80 × 20] = 1,600
3 [80 × 200] = 16,000
4 [80] × 2,000 = 160,000
——
5 [80 × 3 = 2]40
6 [80 × 30] = 2,400
7 [80] × 300 = 24,000
8 [80 × 3,000] = 240,000
——
9 80 × 4 = 3[2]0
10 80 × 40 = 3,200
11 80 × 400 = 32,000
12 80 × 4,000 = 320,000
——
13 80 × 5 = 400
14 80 × 50 = 4,000
15 80 × 500 = 40,000
16 [80] × 5,000 = 400,000
——
_Column 4_
1 [80 × 600] = 48,000
2 [80 × 6,000] = 480,000
——
3 80 × 7 = 5[6]0
4 80 × 70 = 5,600
5 80 × 700 = 5[6],000
6 80 × 7,000 = 5[6]0,000
——
7 80 × 8 = 640
8 80 × 80 = 6,400
9 80 × 800 = 64,000
10 80 × 8,000 = 640,000
——
11 80 × 9 = 720
12 80 × 90 = 7,200
13 80 × 900 = 72,000
14 80 × 9,000 = 720,000
——
15 80 × 10,000 = 80 (should be 800,000)
——
_Column 5_
1 90 × 100 = [9,000]
2 90 × 1,000 = 90,000
——
3 90 × 2 = 180
4 90 × 20 = 1,800
5 90 × 200 = 18,000
6 90 × 2,000 = 180,000
——
7 90 × 3 = 270
8 90 × 30 = 2,700
9 90 × 300 = 27,000
10 90 × 3,000 = 270,000
——
11 90 × 4 = 360
12 90 × 40 = 3,600
13 90 × 400 = 3[6],000
14 90 × 4,000 = 360,000
——
15 90 × 5 = 450
16 90 × 50 = 4,500
17 90 × 500 = 45,000
18 [90 × 5,000] = 4[5]0,000
p. 3 (division by 12, 24, 48)
_Column 1 (÷ 12)_
(top lost: 1–4 ÷ 12)
1 5 (÷ 12 =) ⅓ 1/12
2 6 (÷ 12 =) ½
3 7 (÷ 12 =) ½ 1/12
4 [8] (÷ 12 =) [⅔]
5 9 (÷ 12 =) ½ ¼
6 10 (÷ 12 =) ½ ⅓
7 20 (÷ 12 =) 1⅔
8 30 (÷ 12 =) 2½
9 40 (÷ 12 =) 3⅓
10 50 (÷ 12 =) 4⅙
11 [60] (÷ 12 =) 5
12 70 (÷ 12 =) 5½ [⅓]
13 80 (÷ 12 =) 6⅔
14 90 (÷ 12 =) 7½
15 100 (÷ 12 =) 8⅓
_Column 2 (÷ 12)_
(top lost: 200–800 ÷ 12)
1 900 (÷ 12 =) [75]
2 1,000 (÷ 12 =) 8[3⅓]
3 2,000 (÷ 12 =) [166⅔]
4 3,000 (÷ 12 =) 250
5 4,000 (÷ 12 =) 333[⅓]
6 5,000 (÷ 12 =) 416[⅔]
7 6,000 (÷ 12 =) 500
8 7,000 (÷ 12 =) 583[⅓]
9 8,000 (÷ 12 =) 666⅔
10 9,000 (÷ 12 =) 750
11 10,000 (÷ 12 =) 833[⅓]
(Arabic note after column)
the two qīrāṭs
_Column 3 (÷ 24)_
(top lost: 1–20 ÷ 24)
1 30 (÷ 24 =) 1[¼]
2 40 (÷ 24 =) 1⅔
3 50 (÷ 24 =) 2 1/1[2]
4 60 (÷ 24 =) 2½
5 70 (÷ 24 =) 2½ [⅓ 1/12]
6 80 (÷ 24 =) 3⅓
7 90 (÷ 24 =) 3½ ¼
8 100 (÷ 24 =) 4⅙
_Column 4 (÷ 24)_
(top lost: 200–2000 ÷ 24)
1 3,000 (÷ 24 =) 1[25]
2 4,000 (÷ 24 =) 166⅔
3 5,000 (÷ 24 =) 208[⅓] (?)
4 6,000 (÷ 24 =) 250
5 7,000 (÷ 24 =) 291⅔
6 8,000 (÷ 24 =) 333⅓
7 9,000 (÷ 24 =) 375
8 10,000 (÷ 24 =) 416⅔
(Arabic note after column)
the figures of the qīrāṭ
_Column 5 (÷ 48)_
(top lost: 1–10 ÷ 48)
1 20 (÷ 48 =) [¼ ⅙]
2 30 (÷ 48 =) ½ ⅛
3 40 (÷ 48 =) [½ ⅓]
4 50 (÷ 48 =) 1 [1/24]
5 [60] (÷ 48 =) 1[¼]
6 70 (÷ 48 =) 1[⅓ ⅛]
7 80 (÷ 48 =) 1⅔
8 90 (÷ 48 =) 1½ [⅓ 1/24]
(bottom lost: 100 ÷ 48)
T-S Ar.39.380 1v
p. 1 (multiplication by 50 and 60)
_Column 1_
(top lost: 50 × 5, 50 × 50)
1 [𐋮 𐋷] 𐋢̈ ͵𐋥
2 [𐋮] ͵𐋥 𐋫̈ 𐋥̈
——
3 [𐋮] 𐋦 𐋵
4 𐋮 𐋯 ͵𐋣
5 𐋮 𐋸 𐋣̈
6 𐋮 ͵𐋦 𐋬̈
——
7 𐋮 𐋧 𐋵𐋮
8 𐋮 𐋰 ͵𐋣𐋷
9 𐋮 𐋹 𐋣̈ [͵𐋥]
10 𐋮 ͵𐋧 𐋬̈ 𐋥̈
——
11 𐋮 𐋨 [𐋶]
12 𐋮 𐋱 ͵𐋤
13 𐋮 𐋺 𐋤̈
14 [𐋮] ͵𐋨 𐋭̈
——
_Column 2_
(top lost: 50 × 9, 50 × 90)
1 𐋮 [𐋻 𐋤̈ ͵𐋥]
2 𐋮 ͵[𐋩] 𐋭̈ [𐋥̈]
——
3 𐋮 𐋡̈ 𐋮̈
——
4 𐋯 𐋡 𐋯
5 𐋯 𐋪 𐋸
6 𐋯 𐋳 ͵𐋦
7 𐋯 ͵𐋡 𐋦̈
——
8 𐋯 𐋢 𐋳𐋫
9 𐋯 𐋫 ͵𐋡𐋴
10 𐋯 𐋴 𐋡̈ ͵𐋢
11 𐋯 ͵𐋢 𐋪̈ 𐋢̈
——
12 𐋯 𐋣 𐋳𐋱
13 𐋯 𐋬 ͵𐋡𐋺
14 𐋯 𐋵 𐋡̈ ͵𐋨
15 𐋯 ͵𐋣 𐋪̈ 𐋨̈
——
_Column 3_
(top lost: 60 × 4, 40, 400, 4000)
1 [𐋯 𐋥 𐋵]
2 𐋯 [𐋮 ͵𐋣]
3 𐋯 [𐋷 𐋣̈]
4 𐋯 [͵𐋥 𐋬̈]
——
5 𐋯 [𐋦] 𐋵[𐋯] (?)
6 𐋯 𐋯 [͵𐋣𐋸]
7 𐋯 𐋸 𐋣̈ [͵𐋦]
8 𐋯 ͵𐋦 𐋬̈ 𐋦̈
——
9 𐋯 𐋧 𐋶[𐋫]
10 𐋯 𐋰 ͵𐋤𐋴
11 𐋯 𐋹 𐋤̈ ͵𐋢
12 𐋯 ͵𐋧 𐋭̈ 𐋢̈
——
(Column 4 not preserved except for Arabic note)
.....
رقعة ابن ...
ازهر
والسلم
p. 4 (division by 48 continued)
_Column 1 (÷ 48)_
(top lost: 200–4000 ÷ 48)
1 ͵𐋥 𐋳𐋤 [𐋦ʹ]
2 ͵𐋦 𐋳𐋫𐋥
3 ͵𐋧 𐋳𐋭𐋥 [𐅵 𐋣ʹ]
4 ͵𐋨 𐋳𐋯𐋦 [𐅷]
5 [͵𐋩 𐋳𐋱𐋧 𐅵]
(bottom lost: 10,000 ÷ 48)
p. 1 (multiplication by 50 and 60)
_Column 1_
(top lost: 50 × 5, 50 × 50)
1 [50 × 500] = 25,000
2 [50] × 5,000 = 250,000
——
3 [50] × 6 = 300
4 50 × 60 = 3,000
5 50 × 600 = 30,000
6 50 × 6,000 = 300,000
——
7 50 × 7 = 350
8 50 × 70 = 3,500
9 50 × 700 = 3[5],000
10 50 × 7,000 = 350,000
——
11 50 × 8 = [400]
12 50 × 80 = 4,000
13 50 × 800 = 40,000
14 [50] × 8,000 = 400,000
——
_Column 2_
(top lost: 50 × 9, 50 × 90)
1 50 × [900 = 45,000]
2 50 × [9,000] = 4[5]0,000
——
3 50 × 10,000 = 500,000
——
4 60 × 1 = 60
5 60 × 10 = 600
6 60 × 100 = 6,000
7 60 × 1,000 = 60,000
——
8 60 × 2 = 120
9 60 × 20 = 1,200
10 60 × 200 = 12,000
11 60 × 2,000 = 120,000
——
12 60 × 3 = 180
13 60 × 30 = 1,800
14 60 × 300 = 18,000
15 60 × 3,000 = 180,000
——
_Column 3_
(top lost: 60 × 4, 40, 400, 4000)
1 [60 × 5 = 300]
2 60 × [50 = 3,000]
3 60 × [500 = 30,000]
4 60 × [5,000 = 300,000]
——
5 60 × [6] = 3[6]0 (?)
6 60 × 60 = [3,600]
7 60 × 600 = 3[6],000
8 60 × 6,000 = 360,000
——
9 60 × 7 = 4[2]0
10 60 × 70 = 4,200
11 60 × 700 = 42,000
12 60 × 7,000 = 420,000
——
(Column 4 not preserved except for Arabic note)
.....
Note of Ibn ...
Azhar
Farewell
p. 4 (division by 48 continued)
_Column 1 (÷ 48)_
(top lost: 200–4000 ÷ 48)
1 5,000 (÷ 48 =) 104[⅙]
2 6,000 (÷ 48 =) 125
3 7,000 (÷ 48 =) 145[½ ⅓]
4 8,000 (÷ 48 =) 166[⅔]
5 [9,000] (÷ 48 =) [187½]
(bottom lost: 10,000 ÷ 48)