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Paris Manuscript G — 124

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Bibliothèque de l'Institut de France, Paris

Cono e triangolo, i vertici congiunti da linea punteggiata, con:

seconda - a d - b n c m e

La intera revoluzione della basa colla superfizie del conio, del quale l'assis è equale al diamitro della sua basa, lascia di sé vestigio (sopra loco piano) equale alla superfizie d'esso conio, il quale è un triangolo rettilatero c d

Cono e semicerchio in quattro settori, con:

prima - a - b v c d; f g h - e K i

Se li stremi del lato curvo d'un semicerchio saranno in fra loro in contatto, esso semicerchio si comporrà in figura di conio, e la sua basa sarà un circolo subduplo alla superfizie d'esso conio.

E questa superfizie raddoppiata comporrà la quadratura del cerchio intero, e comporrà un cilindro, del quale la intera revoluzione fatta sopra loco piano lascia di sé vestigio equale a un cerchio, che val 4 volte il cerchio che ha la basa del conio, e vale due volte la superfizie di tal conio, e vale la superfizie d'una spera, che contiene in sé 4 circuli simili alla basa del conio. Adunque direno ch'è il maggior cilindro che del cubo trar si possa1.

Cone and triangle, their vertices joined by a dotted line, with:

second - a d - b n c m e. The complete revolution of the base together with the surface of the cone whose axis is equal to the diameter of its base leaves of itself (upon a flat place) a trace equal to the surface of that cone, which is an equilateral triangle c d.

Cone and semicircle in four sectors, with:

first - a - b v c d; f g h - e K i. If the ends of the curved side of a semicircle are brought into contact with one another, that semicircle will form the figure of a cone, and its base will be a circle half the surface of that cone. And this surface, doubled, will compose the quadrature of the whole circle, and will compose a cylinder whose complete revolution made upon a flat place leaves of itself a trace equal to a circle worth four times the circle that is the cone's base, and worth twice the surface of that cone, and worth the surface of a sphere that contains within itself four circles like the base of the cone. Therefore we shall say it is the largest cylinder that can be drawn from the cube.

Unrolling the cone and squaring the circle

A cone's surface developed as a triangle; a semicircle folded into a cone; the largest cylinder in a cube

Geometry Arithmetic & Proportion

Leonardo continues his cone constructions across two labelled figures. In the 'second' figure (a d, b n c m e) the complete revolution of the base together with the surface of a cone whose axis equals its base diameter leaves upon a plane a trace equal to the cone's surface, an equilateral triangle c d. In the 'first' figure (a b v c d; f g h e K i) a semicircle whose curved ends are brought into contact folds into a cone whose base circle is half its surface. Doubling this composes the quadrature of the whole circle and a cylinder whose rolled trace equals four times the base circle, twice the cone's surface, and the surface of a sphere containing four such circles, declared the largest cylinder drawn from a cube.

On this page

Cone with axis equal to base diameter unrolls as triangle c d

The complete revolution of the base with the surface of the cone whose axis equals its base diameter leaves, on a flat place, a trace equal to the cone's surface, which is an equilateral triangle c d.

Semicircle folded into a cone; the quadrature and largest cylinder

Bringing the ends of a semicircle's curved side into contact forms a cone whose base is a circle half that cone's surface. Doubled, it composes the quadrature of the whole circle and a cylinder whose rolled trace is four times the base circle, twice the cone's surface, and equal to a sphere holding four such circles, said to be the largest cylinder from the cube.

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