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Library/Paris Manuscript M/182

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Paris Manuscript M — 182

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Bibliothèque de l'Institut de France, Paris

The numbers are executed in the non-specular sense, the texts in the specular sense:

1 2 4 8 16 32 64

Because these quantities are in continuous proportion, if you take the first from the last, that is, one from sixty-four, the remainder is sixty-three, which sixty-three is the same as the sum of the units that are found at the middle between one and sixty-four. And it goes like this for any other sum of continuous proportions.

1 3 4 6 8 9 10 12 15 16

These numbers are not placed in any sort of proportion, but if you take the first1 from the last, as you did above, the remainder of the last will be equal to all the excesses that there are from one number to the next. The excess that there is between one and three is two, and from three to four, it is one, which you put above two,2 and that makes three. Proceed in this manner and you will have fourteen.

2 1 2 2 1 1 2 3 1 14

In the outer margin space, executed in the non-specular sense:

2

1

2

2

1

1

2

3

1

14

The numbers are written in ordinary (non-mirror) script, the prose in mirror script

1 2 4 8 16 32 64. Because these quantities are in continuous proportion, if you take the first from the last, that is, one from sixty-four, the remainder is sixty-three, which is the same as the sum of the units found in the middle between one and sixty-four. And it goes like this for any other sum of continuous proportions. 1 3 4 6 8 9 10 12 15 16. These numbers are not placed in any proportion, but if you take the first from the last, as you did above, the remainder of the last will equal all the excesses there are from one number to the next. The excess between one and three is two, and from three to four it is one, which you put above the two, and that makes three. Proceed in this manner and you will have fourteen. 2 1 2 2 1 1 2 3 1 14

In the outer margin, in ordinary (non-mirror) script

2 1 2 2 1 1 2 3 1 14

Continuous Proportions: Summing 1, 2, 4, 8, 16, 32, 64

Last minus first equals the sum of the intermediate terms; a non-proportional row and its differences

Arithmetic & Proportion

The folio compares two rows of numbers. The first, 1 2 4 8 16 32 64, stands in continuous (doubling) proportion, and Leonardo notes that subtracting the first from the last, 64 minus 1 equals 63, which is exactly the sum of the intermediate terms. The second row, 1 3 4 6 8 9 10 12 15 16, is not proportional; here last minus first equals the sum of the successive excesses between neighbouring numbers, which he tallies step by step to fourteen. A column in the outer margin lists those differences, 2 1 2 2 1 1 2 3 1, and their total, 14.

On this page

Geometric series 1 to 64 and the sum of its middle terms

Because the quantities 1 2 4 8 16 32 64 are in continuous proportion, taking the first from the last (one from sixty-four) leaves sixty-three, which equals the sum of the units found in the middle between one and sixty-four. The same holds for any other continuous proportion.

A non-proportional row and its column of differences summing to 14

For the row 1 3 4 6 8 9 10 12 15 16, which is in no proportion, last minus first equals all the excesses from one number to the next: two between one and three, one between three and four, and so on. Added together these give fourteen, listed in the margin as 2 1 2 2 1 1 2 3 1 and totalled 14.

Reader's Edition — described with AI from the e-Leo transcription.