Table arithmétique – Fragment
Notes on the numerical tables of Table arithmétique (Fragment), Latin 9377, f. 113[1] in the Bibliothèque nationale de France (Paris).
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
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 | error: missing leading 6, should be 6975757441 | |
| 6 | 9 | 3 | _ | 4 | 8 | _ | 5 | 7 | 693048057 | 9 x 177 | error: missing leading 3, should be 3693048057 | |
| 9 | 5 | 5 | 1 | 4 | 3 | _ | 8 | 9 | 955143089 | 92 x 176 | error: 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
Then, multiples of 7n (n from 0 to 7) and 13n (n from 8 to 1). Note that exponents add up to 6.
| 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
| 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | transcription (r2l) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|
| . | . | . | . | . | . | . | 9 | 9 | 9 | ||
| . | . | . | . | . | . | 1 | 8 | 81 | 9^2 | ||
| . | . | . | . | . | 9 | 2 | 7 | 729 | 9^3 | ||
| . | . | . | . | 1 | 6 | 5 | 6 | 6561 | 9^4 | ||
| . | . | . | 9 | 4 | 0 | 9 | 5 | 59049 | 9^5 | ||
| . | . | 1 | 8 | 8 | 1 | 3 | 5 | 531881 | 9^6 | ||
| . | 9 | 6 | 9 | 2 | 8 | 7 | 4 | 4782969 | 9^7 | ||
| 1 | 2 | 7 | 6 | 4 | 0 | 3 | 4 | 43046721 | 9^8 | Copulatiu sesquino[vum] et sesquioctava |
Left Side. Column 2, Table 2
| 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | transcription (r2l) | factors | note | -marginalia |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 5 | 7 | 3 | 6 | 2 | 8 | 3 | 38263752 | 8 x 9^7 | ||
| 4 | 2 | 2 | 2 | 1 | 0 | 4 | 3 | 34012224 | 8^2 x 9^6 | ||
| 8 | 8 | 0 | 3 | 3 | 2 | 0 | 3 | 30233088 | 8^3 x 9^5 | ||
| 6 | 5 | 8 | 3 | 7 | 8 | 6 | 2 | 26873856 | 8^4 x 9^4 | ||
| 2 | 7 | 8 | 7 | 8 | 8 | 3 | 2 | 23887872 | 8^5 x 9^3 | ||
| 4 | 6 | 6 | 3 | 3 | 2 | 1 | 2 | 21233664 | 8^6 x 9^2 | ||
| 8 | 6 | 3 | 4 | 7 | 8 | 8 | 1 | 18874368 | 8^7 x 9 | ||
| 6 | 1 | 2 | 7 | 7 | 7 | 6 | 1 | 16777216 | 8^8 | "Copulatiu sesquioctava et sesquiseptiumus" |
Background
The numbers are "apices" of early Arabic numbers.
in a table from "Histoire de la Mathematique" by J.E. Montucla, published in 1757