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Library/Codex Forster I/029v

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Codex Forster I — 029v

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© Victoria and Albert Museum, London

13a

Scambio fra le dimensioni d'una tavola, con:

n m - a o b f - e g - c p d

Una tavola lunga astesa in maggiore lunghezza

sanza mancare di larghe[zza]. Si dimanda

quanto diminuisca della sua prima grossezza.

* La tavola lunga che si propone, bisogna ridurla al suo cilind[r]o1 coll'aiuto della quinta del primo, e poi non avendo essa pella detta quadratura mutata la prima grossezza e avendosi [a] astendere alla data lunghezza sanza diminuizione della larghezza sua, egli è necessario che tale astensione dirivi dalla diminuizione della sua grossezza; per la qual cosa noi astendereno in lungo al dato spazio essa grossezza.

A dire meglio, a b c d sia la lunghezza della tavola, p c sia la misura della sua larghezza, a f sia la data lunghezza, in che la lunghezza della prima tavola s'ha a distendere. Ora avendosi la grossezza e non la larghezza di tal tavola avere mutazione, io quadrerrò essa grossezza di tavola a b c d e ne farò il cilindro n m a o mediante la quinta del primo. Di poi secondo la sesta io farò del quadrato n m a o la grossezza e lunghezza della tavola accresciuta per lunghezza a e f g. Di poi le renderò la sua larghezza, la qual segnai in c p. E così arò fatto il bisogno.

13th.

Exchange between the dimensions of a board, with:

n m - a o b f - e g - c p d. A long board extended to a greater length without lacking in width. It is asked by how much it diminishes from its first thickness. The long board proposed must be reduced to its cylinder with the help of the fifth of the first [book]; and then, since by the said squaring it has not changed its first thickness, and since it must be extended to the given length without diminution of its width, it is necessary that such extension derive from the diminution of its thickness; for which reason we will extend that thickness lengthwise to the given space. To say it better: let a b c d be the length of the board, let p c be the measure of its width, and let a f be the given length to which the length of the first board is to be extended. Now, since the thickness and not the width of such board is to be changed, I will square that thickness of board a b c d and make of it the cylinder n m a o by means of the fifth of the first. Then, according to the sixth, I will make of the square n m a o the thickness and length of the board increased in length a e f g. Then I will give back to it its width, which I marked at c p. And thus I will have done what is needed.

Exchanging the dimensions of a board without losing width

Lengthening a slab by reducing its thickness, and how much thickness is lost

Geometry Arithmetic & Proportion

This leaf treats how to extend a long board (tavola) to a greater length without reducing its width, and asks by how much its thickness must diminish. Leonardo reduces the board to its cylinder by the 'fifth of the first', then argues that because the width is fixed, the added length must come entirely from lost thickness. Working with board a b c d, width p c and target length a f, he squares the thickness, forms the cylinder n m a o, extends it lengthwise to a e f g by the sixth proposition, and finally restores the width marked at c p. A diagram at upper right shows a ribbed box with a swung arc and its construction lines.

On this page

Lengthening a board by reducing thickness only

Let a b c d be the length of the board, p c the measure of its width, and a f the given length to which it is to be extended. Since the thickness, not the width, is to change, Leonardo squares the thickness of board a b c d and makes the cylinder n m a o by the fifth of the first book; by the sixth he stretches the square to the increased length a e f g, then restores the width marked at c p.

Trading thickness for length at constant width

Because the squaring does not change the board's original thickness, and the width must stay fixed, any extension in length has to derive from a diminution of the thickness. Leonardo therefore extends the thickness lengthwise to the given space, so the loss of thickness exactly accounts for the gain in length.

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