Table arithmétique – Fragment
Notes on the numerical tables of Table arithmétique (Fragment), Latin 9377, f. 113[1] in the Bibliothèque nationale de France (Paris).
Other links:
https://gallica.bnf.fr/ark:/12148/btv1b525133302/f238.item
https://archivesetmanuscrits.bnf.fr/ark:/12148/cc77406z
Left side of Latin 9377 f.113
Right side of Latin 9377 f.113
Description of Fragment
The fragment consists of a single sheet of parchment, folded in half, with numerical tables written on both the left and right sides of the inside (flesh-side) of the parchment.
A fascimile of the pages can be viewed at [1]:
outside (reverse of left side)
outside (reverse of right side)
On the left side are 2 columns, each with 3 tables (and each with a cut off table at the bottom. On the right side are 2 columns each with 1 continuous table.
Aspices
0123456789
The Tables
Transcription Notation:
"_" indicates an empty cell in the grid (i.e. the absence of an aspex, what we would represent as a zero).
"." indicates a space (without borders drawn) outside of the grid.
The symbol P with a stroke through its stem in the marginalia is interpreted as the abbreviation for the suffix "per".
Left Side
Left Side, Column 1, Table 1
Ratios of 9 to 17
Multiples of 9n (n from 0 to 7) and 17n (n from 8 to 1). Note that exponents of the two factors add up to 8.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | transcription | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9 | 7 | 5 | 7 | 5 | 7 | 4 | 4 | 1 | 975757441 | 178 | truncated: missing leading 6, should be 6975757441 | |
| 6 | 9 | 3 | _ | 4 | 8 | _ | 5 | 7 | 693048057 | 9 x 177 | truncated: missing leading 3, should be 3693048057 | |
| 9 | 5 | 5 | 1 | 4 | 3 | _ | 8 | 9 | 955143089 | 92 x 176 | truncated: missing leading 1, should be 1955143089 | |
| 1 | 3 | 5 | _ | 7 | 5 | 7 | 5 | 3 | 135075753 | 93 x 175 | error: missing a zero, should be 1035075753 | |
| 5 | 4 | 7 | 9 | 8 | 1 | 2 | 8 | 1 | 547981281 | 94 x 174 | ||
| 2 | 9 | _ | 1 | _ | 7 | 7 | 3 | 7 | 290107737 | 95 x 173 | ||
| 1 | 5 | 3 | 5 | 8 | 6 | 4 | 4 | 9 | 153586449 | 96 x 172 | ||
| _ | 8 | 1 | 3 | 1 | _ | 4 | 7 | 3 | 081310473 | 97 x 17 | "SU(PER) OCTO PARTIENT" |
- ↑ Note that both "9377" and "113" are prime. Coincidence?
Left Side, Column 1, Table 2
Ratios of 16 to 3
First two rows unsolved. Then, multiples of 16n (n from 0 to 5) and 3n (n from 6 to 1). Note that exponents add up to 6.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | transcription | factors | note | -marginalia | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5 | 6 | 2 | 8 | 9 | _ | 6 | 2 | 5 | 562890625 | 58 x 11 x 131 | |||
| 3 | 6 | 6 | 8 | 7 | 5 | . | . | . | 366875 | 54 x 587 | |||
| 7 | 2 | 9 | . | . | . | . | . | . | 729 | 36 | |||
| 3 | 8 | 8 | 8 | . | . | . | . | . | 3888 | 16 x 35 | |||
| 2 | _ | 7 | 3 | 6 | . | . | . | . | 20735 | 162 x 34 | |||
| 1 | 1 | _ | 5 | 9 | 2 | . | . | . | 110592 | 163 x 33 | |||
| . | 5 | 8 | 9 | 8 | 2 | 4 | . | . | 589824 | 164 x 32 | |||
| . | 3 | 1 | 4 | 5 | 7 | 2 | 8 | . | 3145728 | 165 x 3 | "SU(PER) . VII . PARTIENTES" |
Left Side. Column 1, Table 3
Ratios of 7 to 13
Multiples of 7n (n from 0 to 7) and 13n (n from 8 to 1). Note that exponents add up to 8.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | transcription | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 8 | 1 | 5 | 7 | 3 | 0 | 7 | 2 | 1 | 815730721 | 138 | ||
| 4 | 3 | 9 | 2 | 3 | 9 | 6 | 1 | 9 | 439239619 | 7 x 137 | ||
| 2 | 3 | 6 | 5 | 1 | 3 | 6 | 4 | 1 | 236513641 | 72 x 136 | ||
| 1 | 2 | 7 | 3 | 5 | 3 | 4 | 9 | 9 | 127353499 | 73 x 135 | ||
| . | 6 | 8 | 5 | 7 | 4 | 9 | 6 | 1 | 68574961 | 74 x 134 | ||
| . | 3 | 6 | 9 | 2 | 4 | 9 | 7 | 9 | 36924979 | 75 x 133 | ||
| . | 1 | 9 | 8 | 8 | 2 | 6 | 8 | 1 | 19882681 | 76 x 132 | ||
| . | 1 | 0 | ? | 0 | 6 | 0 | 5 | 9 | 10?06059 | 77 x 13 | 4th cell obscures “7”, should be 10706059 | "SU(PER) . VI . PARTIENTES" |
Left Side. Column 1, Table 4
cut off after 1 row.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | transcription | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ? | 4 | 3 | 5 | 8 | 8 | 0 | 7 | 6 | ?43588076 | ? |
Left Side. Column 2, Table 1
Powers of 9
| 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | transcription (r2l) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|
| . | . | . | . | . | . | . | 9 | 9 | 9 | ||
| . | . | . | . | . | . | 1 | 8 | 81 | 92 | ||
| . | . | . | . | . | 9 | 2 | 7 | 729 | 93 | ||
| . | . | . | . | 1 | 6 | 5 | 6 | 6561 | 94 | ||
| . | . | . | 9 | 4 | 0 | 9 | 5 | 59049 | 95 | ||
| . | . | 1 | 8 | 8 | 1 | 3 | 5 | 531881 | 96 | ||
| . | 9 | 6 | 9 | 2 | 8 | 7 | 4 | 4782969 | 97 | ||
| 1 | 2 | 7 | 6 | 4 | 0 | 3 | 4 | 43046721 | 98 | "Copulatiu sesquino[vum] et sesquioctava" |
Left Side. Column 2, Table 2
Ratios of 8 to 9
Multiples of 8n (n from 1 to 8) and 9n (n from 7 to 0). Note that exponents add up to 8.
| 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | transcription (r2l) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 5 | 7 | 3 | 6 | 2 | 8 | 3 | 38263752 | 8 x 97 | ||
| 4 | 2 | 2 | 2 | 1 | 0 | 4 | 3 | 34012224 | 82 x 96 | ||
| 8 | 8 | 0 | 3 | 3 | 2 | 0 | 3 | 30233088 | 83 x 95 | ||
| 6 | 5 | 8 | 3 | 7 | 8 | 6 | 2 | 26873856 | 84 x 94 | ||
| 2 | 7 | 8 | 7 | 8 | 8 | 3 | 2 | 23887872 | 85 x 93 | ||
| 4 | 6 | 6 | 3 | 3 | 2 | 1 | 2 | 21233664 | 86 x 92 | ||
| 8 | 6 | 3 | 4 | 7 | 8 | 8 | 1 | 18874368 | 87 x 9 | ||
| 6 | 1 | 2 | 7 | 7 | 7 | 6 | 1 | 16777216 | 88 | "Copulatiu sesquioctava et sesquiseptiumus" |
Left Side. Column 2, Table 3
Ratios of 8 to 7
Multiples of 8n (n from 7 to 0) and 7n (n from 1 to 8). Note that exponents add up to 8.
| 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | transcription (r2l) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 4 | 6 | 0 | 0 | 8 | 6 | 4 | 1 | 14680064 | 87 x 7 | ||
| 6 | 5 | 0 | 5 | 4 | 8 | 2 | 1 | 12845056 | 86 x 72 | ||
| 4 | 2 | 4 | 9 | 3 | 2 | 1 | 1 | 11239424 | 85 x 73 | ||
| 6 | 9 | 4 | 4 | 3 | 8 | 9 | . | 9834496 | 84 x 74 | ||
| 4 | 8 | 1 | 5 | 0 | 6 | 8 | . | 8605184 | 83 x 75 | ||
| 6 | 3 | 5 | 9 | 2 | 5 | 7 | . | 7529536 | 82 x 76 | ||
| 4 | 4 | 3 | 8 | 8 | 5 | 6 | . | 6588344 | 8 x 77 | ||
| 1 | 0 | 8 | 4 | 6 | 7 | 5 | . | 5764801 | 78 | "Copulatiai sesquiseptum et sesquisexta" |
Right Side
Right Side, Column 1, Rows 1-9
Powers of 9
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | transcription (l2r) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9 | . | . | . | . | . | . | . | . | 9 | 9 | ||
| 8 | 1 | . | . | . | . | . | . | . | 81 | 92 | ||
| 7 | 2 | 9 | . | . | . | . | . | . | 729 | 93 | ||
| 6 | 5 | 6 | 1 | . | . | . | . | . | 6561 | 94 | ||
| 5 | 9 | _ | 4 | 9 | . | . | . | . | 59049 | 95 | ||
| 5 | 3 | 1 | 4 | 4 | 1 | . | . | . | 531441 | 96 | ||
| 4 | 7 | 8 | 2 | 9 | 6 | 9 | . | . | 4782969 | 97 | ||
| 4 | 3 | _ | 4 | 6 | 7 | 2 | 1 | . | 43046721 | 98 | ||
| 3 | 8 | 7 | 4 | 2 | _ | 4 | 8 | 9 | 387420489 | 99 | "SU(PER) OCTO PARTIENTES" |
Right Side, Column 1, Rows 10-18
Ratios of 8 to 9
Multiples of 8n (n from 1 to 9) and 9n (n from 8 to 0). Note that exponents add up to 9.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | transcription (l2r) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3 | 4 | 4 | 3 | 7 | 3 | 7 | 6 | 8 | 344373768 | 8 x 98 | ||
| 3 | _ | 6 | 1 | 1 | _ | _ | 1 | 6 | 306110016 | 82 x 97 | ||
| 2 | 7 | 2 | _ | 9 | 7 | 7 | 9 | 2 | 272097792 | 83 x 96 | ||
| 2 | 4 | 1 | 8 | 6 | 4 | 7 | _ | 4 | 241864704 | 84 x 95 | ||
| 2 | 1 | 4 | 9 | 9 | _ | 8 | 4 | 8 | 214990848 | 85 x 94 | ||
| 1 | 9 | 1 | 1 | _ | 2 | 9 | 7 | 6 | 191102976 | 86 x 93 | ||
| 1 | 6 | 9 | 8 | 6 | 9 | 3 | 1 | 2 | 169869312 | 87 x 92 | ||
| 1 | 5 | _ | 9 | 9 | 4 | 9 | 4 | 4 | 150994944 | 88 x 9 | ||
| 1 | 3 | 4 | 2 | 1 | 7 | 7 | 2 | 8 | 134217728 | 89 | "SU(PER) 7 PARTIENTES" |
Right Side, Column 1, Rows 19-27
Ratios of 8 to 7
Multiples of 8n (n from 8 to 0) and 7n (n from 1 to 9). Note that exponents add up to 9.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | transcription (l2r) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 7 | 4 | 4 | _ | 5 | 1 | 2 | 117440512 | 88 x 7 | ||
| 1 | _ | 2 | 7 | 6 | _ | 4 | 4 | 8 | 102760448 | 87 x 72 | ||
| . | 8 | 9 | 9 | 1 | 5 | 3 | 9 | 2 | 89915392 | 86 x 73 | ||
| . | 7 | 8 | 6 | 7 | 5 | 9 | 6 | 8 | 78675968 | 85 x 74 | ||
| . | 6 | 8 | 8 | 4 | 1 | 4 | 7 | 2 | 68841472 | 84 x 75 | ||
| . | 6 | 2 | 3 | 6 | 2 | 8 | 8 | 60236288 | 83 x 76 | |||
| . | 5 | 2 | 7 | 6 | 7 | 5 | 2 | 52706752 | 82 x 77 | |||
| . | 4 | 6 | 1 | 1 | 8 | 4 | 8 | 46118408 | 8 x 78 | "SU(PER) 6 PARTIENTES" | ||
| . | 4 | 3 | 5 | 3 | 6 | 7 | 40353607 | 79 |
Right Side, Column 2, Rows 1-9
Ratios of 17 to 9
Multiples of 17n (n from 9 to 1) and 9n (n from 0 to 8). Note that exponents add up to 9.
| 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | transcription (r2l) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7 | 9 | 4 | 6 | 7 | 8 | 7 | 8 | 5 | 8 | 1 | 1 | 118587876497 | 179 | ||
| 9 | 6 | 9 | 6 | 1 | 8 | 1 | 8 | 7 | 2 | 6 | . | 62781816969 | 178 x 9 | ||
| 3 | 1 | 5 | 2 | 3 | 4 | 7 | 3 | 2 | 3 | 3 | . | 33237432513 | 177 x 92 | ||
| 1 | _ | 8 | 7 | 8 | 2 | 6 | 9 | 5 | 7 | 1 | . | 17596287801 | 176 x 93 | ||
| 7 | 7 | 7 | 1 | 8 | 6 | 5 | 1 | 3 | 9 | . | . | 9315681777 | 175 x 94 | ||
| 9 | 2 | 5 | 1 | 3 | 8 | 1 | 3 | 9 | 4 | . | . | 4931831529 | 174 x 95 | ||
| 3 | 3 | 6 | 9 | 6 | 9 | _ | 1 | 6 | 2 | . | . | 2610969633 | 173 x 96 | ||
| 1 | 4 | _ | 8 | 7 | 2 | 2 | 8 | 3 | 1 | . | . | 1382278041 | 172 x 97 | ||
| 7 | 5 | 2 | 4 | 9 | 7 | 1 | 3 | 7 | ( ) | . | . | 731794257 | 17 x 98 |
Right Side, Column 2, Rows 10-12
Ratios of 9 to 5?
| 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | transcription (r2l) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5 | 7 | 3 | 9 | 5 | 3 | 3 | 4 | 4 | 8 | 3 | . | 38443359375 | 97 x 59 | ||
| . | . | . | 5 | 2 | 1 | 3 | _ | 5 | _ | 2 | . | 20503125 | 96 x 55 | ||
| . | . | . | . | . | . | 5 | 3 | 9 | _ | 1 | . | 10935 | 95 x 5 |
Right Side, Column 2, Rows 13-18
Ratios of 16 to 3?
Multiples of 2n (n = 3,7,11,15,19) and 3n (n= 6 to 1). The product increases 16x (24) between rows.
| 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | transcription (r2l) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| . | . | . | . | . | . | 2 | 3 | 8 | 5 | . | . | 5832 | 23 x 36 | ||
| . | . | . | . | . | 4 | _ | 1 | 1 | 3 | . | . | 31104 | 27 x 35 | ||
| . | . | . | . | 8 | 8 | 8 | 5 | 6 | 1 | . | . | 165888 | 211 x 34 | ||
| . | . | . | 6 | 3 | 7 | 4 | 8 | 8 | . | . | . | 884736 | 215 x 33 | ||
| . | . | 2 | 9 | 5 | 8 | 1 | 7 | 4 | . | . | . | 4718592 | 219 x 32 | ||
| . | 4 | 2 | 8 | 5 | 6 | 1 | 5 | 2 | . | . | . | 25165824 | 223 x 3 |
Right Side, Column 2, Rows 19-27
Ratios of 13 to 7
Multiples of 13n (n from 9 to 1) and 7n (n from 0 to 8). Note that exponents add up to 9.
| 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | transcription (r2l) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3 | 7 | 3 | 9 | 9 | 4 | 4 | _ | 6 | . | 1 | . | 10604499373 | 139 | ||
| 7 | 4 | _ | 5 | 1 | 1 | _ | 1 | 7 | 5 | . | . | 5710115047 | 138 x 7 | ||
| 3 | 3 | 3 | 7 | 7 | 6 | 4 | 7 | _ | 3 | . | . | 3074677333 | 137 x 72 | ||
| 7 | 8 | 4 | 5 | 9 | 5 | 5 | 5 | 6 | 1 | . | . | 1655595487 | 136 x 73 | ||
| 3 | 9 | 4 | 4 | 7 | 4 | 1 | 9 | 8 | . | . | . | 891474493 | 135 x 74 | ||
| 7 | 2 | 7 | 4 | 2 | _ | _ | 8 | 4 | . | . | . | 480024727 | 134 x 75 | ||
| 3 | 5 | 8 | 4 | 7 | 4 | 8 | 5 | 2 | . | . | . | 258474853 | 133 x 76 | ||
| 7 | 6 | 7 | 8 | 7 | 1 | 9 | 3 | 1 | . | . | . | 139178767 | 132 x 177 | ||
| 3 | 1 | 4 | 2 | 4 | 9 | 4 | 7 | ( ) | . | . | . | 74942413 | 13 x 78 |
Right Side, Column 2, Rows 28-29
Ratios of 11 (cut off)
| 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | transcription (r2l) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 9 | 6 | 7 | 4 | 9 | 7 | 5 | 3 | 2 | . | . | 2357947691 | 119 | ||
| 6 | 8 | 2 | 3 | 5 | 1 | 6 | 8 | 2 | 1 | . | . | 1286153286 | 2 x 3 x 118 |
Background
The numbers are "apices" of early Arabic numbers.
in a table from "Histoire de la Mathematique" by J.E. Montucla, published in 1757