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Paris Manuscript F — 117

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Bibliothèque de l'Institut de France, Paris

Quattro figure geometriche, con:

c - b e - a f d - K L; i - g h - m - n; a

Quadrami il quadrangolo a.

Provasi che 'l mezzo cilindro colla quinta si po transmutare di punto nel mezzo cubo sanza accrescimento o diminuizione di se medesimo. El simile si po fare del mezzo cubo nel mezzo cilindro.

K L mezzo cubo è equale al m n mezzo cilindro. Restaci1 a provare che 'l solido dell'angolo che sta di là alle 2 dette superfizie, sieno equali per far tacere l'avversario, che dice non essere e[q]uali. g h, basa del triangolo g i h, è posto equale alla basa a f, e similmente alla basa f d, e l'angolo di ciascuno è posto retto. Seguita per la ottava che tutto il triangolo a c d, avendo la basa dupla alla basa g h, esso triangolo a b c è quadruplo al triangolo g i h. Seguita essere più il doppio che non si conviene2.

Figura di mistura3:

Se farai cannelle a uso di penne d'oca, le quali sieno fisse4 e bianche e vestite dentro di nero e poi di transparente, e di fori di sardonio5 e poi di transparente, e tutta la grossezza di tal canne sia composta di queste tali misture. Poi le 'numidisci e serrale e lascia seccare nello strettoio, il quale6 se le serri per piano, faranno un effetto; se le serri in quadrato, ne faranno un altro, e 'l simile se le serri in tria[n]golo. Ma se tu le serri per fronte o avviluppate per diversi aspetti, ancora farà bene.

Ma se nel transparente ispessato al sole tu con piccolo stile farai mistione di diversi colori, massime nero e bianco fisso e giallo d'orpimento7 bruciato, tu farai bellissimi retrosi e varie macchie minute e liniate a uso d'agatis8.

Four geometric figures, with:

c - b e - a f d - K L; i - g h - m - n; a. Square the quadrangle a. It is proved that the half-cylinder, together with the fifth, can be transmuted point for point into the half-cube without any increase or diminution of itself. The same can be done with the half-cube into the half-cylinder. K L, the half-cube, is equal to m n, the half-cylinder. It remains for us to prove that the solid of the angle lying beyond the two said surfaces are equal, so as to silence the adversary who says they are not equal. g h, the base of the triangle g i h, is set equal to the base a f, and likewise to the base f d, and the angle of each is set as a right angle. It follows, by the eighth, that the whole triangle a c d, having its base double the base g h, that triangle a b c is quadruple the triangle g i h. It follows that it is more than double what is fitting.

Figure of mixture:

If you make little tubes in the fashion of goose quills, which are fixed and white and lined inside with black and then with transparent, and outside with sardonyx and then with transparent, and the whole thickness of such tubes is composed of these mixtures. Then moisten them, close them, and let them dry in the press; which, if it presses them flat, they will make one effect; if you press them square, they will make another; and likewise if you press them in a triangle. But if you press them end-on or wrapped in various aspects, it will still turn out well. But if in the transparent thickened in the sun you make, with a small stylus, a mixing of various colours, especially black and fixed white and yellow of burnt orpiment, you will make most beautiful whorls and various minute spots streaked in the manner of agate.

A half-cylinder into a half-cube, and an agate recipe

A solid-geometry transmutation proof, and quill-tubes layered to imitate agate

Geometry Ornament & Heraldry

The main column is a solid-geometry proof that a half-cylinder can be transmuted point for point into a half-cube of the same volume, and vice versa, arguing over the equality of the corner solids by comparing bases and triangles (K L, m n, a f, g h, a c d, g i h). Small figures at the upper right show a divided prism, a house-shaped solid, a rhombus and a hexagonal 'quadrangle'. A second passage, illustrated by a marbled blob, gives a craft recipe for making quill-like tubes layered with black, white, sardonyx and transparent to produce whorled, agate-like patterns.

On this page

Transmuting a half-cylinder into a half-cube (K L, m n)

Leonardo claims the half-cylinder, with the fifth part, can be transmuted point for point into the half-cube without change of volume, and the reverse. The half-cube K L is set equal to the half-cylinder m n; it then remains to prove the corner solids equal, arguing that triangle a c d, with base double base g h, is quadruple triangle g i h.

Layered quill-tubes that imitate agate

Make little tubes like goose quills, white and lined within with black then transparent, and outside with sardonyx then transparent, and press them flat, square or triangular for different effects. Working various colours into the sun-thickened transparent with a small stylus produces beautiful whorls and minute streaked spots in the manner of agate.

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