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Library/Codex Forster I/018r

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Codex Forster I — 018r

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© Victoria and Albert Museum, London

'Linee giudiziali' comprendenti quattro coppie di solidi, con:

K - q r grossezza - o p larghezza - n m - n cubo, m tavola cubata e po' cilindrata - i g x - S t lunghezza - L v h

A sinistra. Cubo, con:

d

____________ a lunghezza

________ b larghezza

___ c grossezza

D'un cubo dato si facci una tavola con lunghezza, larghezza e grossezza proporzionata inrazionale, qual è quella di 3 linie date in fra loro inrazionali.

Volta carta e vedra' meglio.

Metti la data proporzione delle 3 linie a b c in forma di tavola, di poi colla quinta del primo farai d'essa tavola il suo cilindro, poi colla prima del secondo farai d'esso cilindro il suo cubo. Fatto questo, colla terza del secondo farai del cubo un cilindro d'una data gr[o]ssezza che sia equale al primo dato cubo d, che sarà n m, de' quali n è il primo dato cubo, m fu la tavola nata della proporzione inrazionale delle 3 date linie a b c, e poi fu cubata e poi cilindrata in esso cilindro m, ch'è di grossezza equale al cubo n. Ora noi vediamo la proporzione che è dal primo dato cubo alla tavola composta delle 3 date linie a b c, cioè la proporzione i g con g x, onde noi produrreno le 3 linie1.

'Judicial lines' comprising four pairs of solids, with:

K — q r thickness — o p width — n m — n cube, m slab cubed and then cylindered — i g x — S t length — L v h

On the left. Cube, with:

d

—— a length

— b width

— c thickness

From a given cube let a slab be made with length, width and thickness in irrational proportion, such as that of 3 given mutually irrational lines.

Turn the page and you will see better.

Set the given proportion of the 3 lines a b c in the form of a slab; then with the fifth of the first you will make of this slab its cylinder; then with the first of the second you will make of this cylinder its cube. This done, with the third of the second you will make of the cube a cylinder of a given thickness that is equal to the first given cube d, which will be n m, of which n is the first given cube, m was the slab born of the irrational proportion of the 3 given lines a b c, and was then cubed and then cylindered into that cylinder m, which is of thickness equal to the cube n. Now we see the proportion that is from the first given cube to the slab composed of the 3 given lines a b c, that is the proportion i g with g x, whence we shall produce the 3 lines.

Turning a Cube into an Irrationally-Proportioned Slab

Stereometry: converting solids of equal volume through slab, cylinder and cube

Geometry Arithmetic & Proportion

The page treats the transformation of solids of equal volume: a given cube d is to be reworked into a slab whose length a, width b and thickness c stand in irrational proportion, like that of three mutually irrational lines. Leonardo lays out the procedure step by step, passing from slab to cylinder to cube by citing numbered propositions ('the fifth of the first', 'the third of the second'). A vertical column of four stacked solid-pairs at right, marked with 'judicial lines' (n the cube, m the slab), illustrates the successive conversions, and a note reads 'turn the page and you will see better.'

On this page

From a given cube to a slab of irrational proportion

Leonardo poses the problem: from a given cube d, make a slab whose length a, width b and thickness c are in irrational proportion, like that of three mutually irrational lines. He solves it by setting the three lines as a slab, then forming its cylinder, then its cube, then a cylinder of given thickness equal to the original cube n.

Column of four solid-pairs ('judicial lines')

A vertical stack at right pairs the successive solids labelled by letters. n is the first given cube; m is the slab born of the irrational proportion of the three lines a b c, cubed and then rendered as a cylinder equal in thickness to cube n.

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