Difference between revisions of "Table arithmétique – Fragment"

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(added Left column 2 table 3)
(→‎The Tables: added right side column 1)
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==The Tables==
 
==The Tables==
  
−
===Left Side, Column 1, Table 1===
+
===Left Side===
 +
====Left Side, Column 1, Table 1====
  
 
'''Ratios of 9 to 17'''
 
'''Ratios of 9 to 17'''
Line 51: Line 52:
 
</tab>
 
</tab>
  
−
===Left Side, Column 1, Table 2===
+
====Left Side, Column 1, Table 2====
  
 
'''Ratios of 16 to 3'''
 
'''Ratios of 16 to 3'''
Line 69: Line 70:
 
</tab>
 
</tab>
  
−
===Left Side. Column 1, Table 3===
+
====Left Side. Column 1, Table 3====
  
 
'''Ratios of 7 to 13'''
 
'''Ratios of 7 to 13'''
Line 81: Line 82:
 
2 3 6 5 1 3 6 4 1 236513641 7<sup>2</sup> x 13<sup>6</sup>
 
2 3 6 5 1 3 6 4 1 236513641 7<sup>2</sup> x 13<sup>6</sup>
 
1 2 7 3 5 3 4 9 9 127353499 7<sup>3</sup> x 13<sup>5</sup>
 
1 2 7 3 5 3 4 9 9 127353499 7<sup>3</sup> x 13<sup>5</sup>
−
6 8 5 7 4 9 6 1 68574961 7<sup>4</sup> x 13<sup>4</sup>
+
. 6 8 5 7 4 9 6 1 68574961 7<sup>4</sup> x 13<sup>4</sup>
−
3 6 9 2 4 9 7 9 36924979 7<sup>5</sup> x 13<sup>3</sup>
+
. 3 6 9 2 4 9 7 9 36924979 7<sup>5</sup> x 13<sup>3</sup>
−
1 9 8 8 2 6 8 1 19882681 7<sup>6</sup> x 13<sup>2</sup>
+
. 1 9 8 8 2 6 8 1 19882681 7<sup>6</sup> x 13<sup>2</sup>
−
1 0 ? 0 6 0 5 9 10?06059 7<sup>7</sup> x 13 4th cell obscures “7”, should be 10706059 "SU(PER) . VI . PARTIENTES"
+
. 1 0 ? 0 6 0 5 9 10?06059 7<sup>7</sup> x 13 4th cell obscures “7”, should be 10706059 "SU(PER) . VI . PARTIENTES"
 
</tab>
 
</tab>
  
−
===Left Side. Column 1, Table 4 ===
+
====Left Side. Column 1, Table 4 ====
  
 
cut off after 1 row.
 
cut off after 1 row.
Line 96: Line 97:
 
</tab>
 
</tab>
  
−
===Left Side. Column 2, Table 1 ===
+
====Left Side. Column 2, Table 1 ====
  
 
'''Powers of 9'''
 
'''Powers of 9'''
Line 103: Line 104:
 
8 7 6 5 4 3 2 1 transcription (r2l) factors note marginalia
 
8 7 6 5 4 3 2 1 transcription (r2l) factors note marginalia
 
. . . . . . . 9 9 9
 
. . . . . . . 9 9 9
−
. . . . . . 1 8 81 9^2
+
. . . . . . 1 8 81 9<sup>2</sup>
−
. . . . . 9 2 7 729 9^3
+
. . . . . 9 2 7 729 9<sup>3</sup>
−
. . . . 1 6 5 6 6561 9^4
+
. . . . 1 6 5 6 6561 9<sup>4</sup>
−
. . . 9 4 0 9 5 59049 9^5
+
. . . 9 4 0 9 5 59049 9<sup>5</sup>
−
. . 1 8 8 1 3 5 531881 9^6
+
. . 1 8 8 1 3 5 531881 9<sup>6</sup>
−
. 9 6 9 2 8 7 4 4782969 9^7
+
. 9 6 9 2 8 7 4 4782969 9<sup>7</sup>
−
1 2 7 6 4 0 3 4 43046721 9^8 "Copulatiu sesquino[vum] et sesquioctava"
+
1 2 7 6 4 0 3 4 43046721 9<sup>8</sup> "Copulatiu sesquino[vum] et sesquioctava"
 
</tab>
 
</tab>
  
−
===Left Side. Column 2, Table 2 ===
+
====Left Side. Column 2, Table 2 ====
  
 
'''Ratios of 8 to 9'''
 
'''Ratios of 8 to 9'''
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<tab sep=tab class="wikitable" border = "1" cellspacing="3" cellpadding = "3" head = "top>
 
<tab sep=tab class="wikitable" border = "1" cellspacing="3" cellpadding = "3" head = "top>
 
8 7 6 5 4 3 2 1 transcription (r2l) factors note marginalia
 
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
+
2 5 7 3 6 2 8 3 38263752 8 x 9<sup>7</sup>
−
4 2 2 2 1 0 4 3 34012224 8^2 x 9^6
+
4 2 2 2 1 0 4 3 34012224 8<sup>2</sup> x 9<sup>6</sup>
−
8 8 0 3 3 2 0 3 30233088 8^3 x 9^5
+
8 8 0 3 3 2 0 3 30233088 8<sup>3</sup> x 9<sup>5</sup>
−
6 5 8 3 7 8 6 2 26873856 8^4 x 9^4
+
6 5 8 3 7 8 6 2 26873856 8<sup>4</sup> x 9<sup>4</sup>
−
2 7 8 7 8 8 3 2 23887872 8^5 x 9^3
+
2 7 8 7 8 8 3 2 23887872 8<sup>5</sup> x 9<sup>3</sup>
−
4 6 6 3 3 2 1 2 21233664 8^6 x 9^2
+
4 6 6 3 3 2 1 2 21233664 8<sup>6</sup> x 9<sup>2</sup>
−
8 6 3 4 7 8 8 1 18874368 8^7 x 9
+
8 6 3 4 7 8 8 1 18874368 8<sup>7</sup> x 9
−
6 1 2 7 7 7 6 1 16777216 8^8 "Copulatiu sesquioctava et sesquiseptiumus"
+
6 1 2 7 7 7 6 1 16777216 8<sup>8</sup> "Copulatiu sesquioctava et sesquiseptiumus"
 
</tab>
 
</tab>
  
−
===Left Side. Column 2, Table 3 ===
+
====Left Side. Column 2, Table 3 ====
  
 
'''Ratios of 8 to 7'''
 
'''Ratios of 8 to 7'''
Line 134: Line 135:
 
<tab sep=tab class="wikitable" border = "1" cellspacing="3" cellpadding = "3" head = "top>
 
<tab sep=tab class="wikitable" border = "1" cellspacing="3" cellpadding = "3" head = "top>
 
8 7 6 5 4 3 2 1 transcription (r2l) factors note marginalia
 
8 7 6 5 4 3 2 1 transcription (r2l) factors note marginalia
−
4 6 0 0 8 6 4 1 14680064 8^7 x 7
+
4 6 0 0 8 6 4 1 14680064 8<sup>7</sup> x 7
−
6 5 0 5 4 8 2 1 12845056 8^6 x 7^2
+
6 5 0 5 4 8 2 1 12845056 8<sup>6</sup> x 7<sup>2</sup>
−
4 2 4 9 3 2 1 1 11239424 8^5 x 7^3
+
4 2 4 9 3 2 1 1 11239424 8<sup>5</sup> x 7<sup>3</sup>
−
6 9 4 4 3 8 9 . 9834496 8^4 x 7^4
+
6 9 4 4 3 8 9 . 9834496 8<sup>4</sup> x 7<sup>4</sup>
−
4 8 1 5 0 6 8 . 8605184 8^3 x 7^5
+
4 8 1 5 0 6 8 . 8605184 8<sup>3</sup> x 7<sup>5</sup>
−
6 3 5 9 2 5 7 . 7529536 8^2 x 7^6
+
6 3 5 9 2 5 7 . 7529536 8<sup>2</sup> x 7<sup>6</sup>
−
4 4 3 8 8 5 6 . 6588344 8 x 7^7
+
4 4 3 8 8 5 6 . 6588344 8 x 7<sup>7</sup>
−
1 0 8 4 6 7 5 . 5764801 7^8 "Copulatiai sesquiseptum et sesquisexta"
+
1 0 8 4 6 7 5 . 5764801 7<sup>8</sup> "Copulatiai sesquiseptum et sesquisexta"
 
</tab>
 
</tab>
 +
 +
===Right Side===
 +
 +
====Right Side, Column 1, Rows 1-9====
 +
 +
'''Powers of 9'''
 +
 +
<tab sep=tab class="wikitable" border = "1" cellspacing="3" cellpadding = "3" head = "top>
 +
9 . . . . . . . . 9 9
 +
8 1 . . . . . . . 81 9<sup>2</sup>
 +
7 2 9 . . . . . . 729 9<sup>3</sup>
 +
6 5 6 1 . . . . . 6561 9<sup>4</sup>
 +
5 9 _ 4 9 . . . . 59049 9<sup>5</sup>
 +
5 3 1 4 4 1 . . . 531441 9<sup>6</sup>
 +
4 7 8 2 9 6 9 . . 4782969 9<sup>7</sup>
 +
4 3 _ 4 6 7 2 1 . 43046721 9<sup>8</sup>
 +
3 8 7 4 2 _ 4 8 9 387420489 9<sup>9</sup> "SU(PER) OCTO PARTIENTES"
 +
</tab>
 +
 +
====Right Side, Column 1, Rows 10-18====
 +
 +
'''Ratios of 8 to 9'''
 +
 +
<tab sep=tab class="wikitable" border = "1" cellspacing="3" cellpadding = "3" head = "top>
 +
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 9<sup>8</sup>
 +
3 _ 6 1 1 _ _ 1 6 306110016 8<sup>2</sup> x 9<sup>7</sup>
 +
2 7 2 _ 9 7 7 9 2 272097792 8<sup>3</sup> x 9<sup>6</sup>
 +
2 4 1 8 6 4 7 _ 4 241864704 8<sup>4</sup> x 9<sup>5</sup>
 +
2 1 4 9 9 _ 8 4 8 214990848 8<sup>5</sup> x 9<sup>4</sup>
 +
1 9 1 1 _ 2 9 7 6 191102976 8<sup>6</sup> x 9<sup>3</sup>
 +
1 6 9 8 6 9 3 1 2 169869312 8<sup>7</sup> x 9<sup>2</sup>
 +
1 5 _ 9 9 4 9 4 4 150994944 8<sup>8</sup> x 9
 +
1 3 4 2 1 7 7 2 8 134217728 8<sup>9</sup> "SU(PER) 7 PARTIENTES"
 +
</tab>
 +
 +
====Right Side, Column 1, Rows 19-27====
 +
 +
'''Ratios of 8 to 7'''
 +
<tab sep=tab class="wikitable" border = "1" cellspacing="3" cellpadding = "3" head = "top>
 +
1 2 3 4 5 6 7 8 9 transcription (l2r) factors note marginalia
 +
1 1 7 4 4 _ 5 1 2 117440512 8<sup>8</sup> x 7
 +
1 _ 2 7 6 _ 4 4 8 102760448 8<sup>7</sup> x 7<sup>2</sup>
 +
. 8 9 9 1 5 3 9 2 89915392 8<sup>6</sup> x 7<sup>3</sup>
 +
. 7 8 6 7 5 9 6 8 78675968 8<sup>5</sup> x 7<sup>4</sup>
 +
. 6 8 8 4 1 4 7 2 68841472 8<sup>4</sup> x 7<sup>5</sup>
 +
. 6 2 3 6 2 8 8 60236288 8<sup>3</sup> x 7<sup>6</sup>
 +
. 5 2 7 6 7 5 2 52706752 8<sup>2</sup> x 7<sup>7</sup>
 +
. 4 6 1 1 8 4 8 46118408 8 x 7 ^ 8 "SU(PER) 6 PARTIENTES"
 +
. 4 3 5 3 6 7 40353607 7<sup>9</sup>
 +
</tab>
 +
 
==Background==
 
==Background==
  

Revision as of 15:22, 2 August 2022

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)

left side

right 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

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"
  1. ↑ 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

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

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

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

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

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

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 7 ^ 8 "SU(PER) 6 PARTIENTES"
. 4 3 5 3 6 7 40353607 79

Background

The numbers are "apices" of early Arabic numbers.

in a table from "Histoire de la Mathematique" by J.E. Montucla, published in 1757


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