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
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 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | SUE OCTO PARTIENT |
- ↑ Note that both "9377" and "113" are prime. Coincidence?
Left Side, Column 1, Table 2
First two roles 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | SUE 7 PARTIENTES |
Left Side. Column 1, Table 3
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 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 be10706059 |
Background
The numbers are "apices" of early Arabic numbers.
in a table from "Histoire de la Mathematique" by J.E. Montucla, published in 1757