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Paris Manuscript F — 127

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Bibliothèque de l'Institut de France, Paris

Prespettiva.

Due linee parallele, con:

a t x e b; (c f) g f K

De le cose remosse dall'occhio [con] equale distanzia, quella parrà esser men diminuita che prima era più1.

Delle cose remosse dall'occhio con equal distanzia dal lor primo sito quella men diminuisce, che prima era più distante da esso occhio. E tal fia la proporzione della diminuizione, qual fu la proporzione delle distanzie, ch'esse avean da l'occhio avanti il loro moto.

Come dire: il corpo t e 'l corpo e che è che2 la proporzion delle lor distanzie dall'occhio a è quincupla. Io removo ciascun del suo sito e lo fo più distante dall'occhio uno d'essi quinti in ch'è divisa la proporzione. Accade dunque che [il corpo] più vicino all'occhio ha raddoppiata la distanzia e per la penultima di questo esso è diminu[i]to la metà del suo tutto, e 'l corpo e per lo medesimo moto è diminuito 1/5 d'esso suo tutto. Adunque per la detta penultima è vero quel che in questa pe[n]ultima s'è preposto; e questo dico per li moti de' corpi celesti, che 3500 miglia di distanzia, ch'elli han più essendo in oriente che sopra di noi, non crescano o diminuiscano con sensibile dimostrazione.

Perspective.

Two parallel lines, with:

a t x e b; (c f) g f K

Of things removed from the eye by an equal distance, the one that was formerly farther will appear less diminished than before.

Of things removed from the eye by an equal distance from their first position, the one that was previously more distant from that eye diminishes less. And the proportion of the diminution will be as the proportion of the distances they had from the eye before their movement.

That is to say: take the body t and the body e, whose distances from the eye a stand in a quintuple proportion. I remove each from its position and make it more distant from the eye by one of those fifths into which the proportion is divided. It happens then that the body nearer the eye has doubled its distance, and by the penultimate [proposition] of this it is diminished by half of its whole; and the body e, by the same movement, is diminished by 1/5 of its whole. Therefore, by the said penultimate, what has been proposed in this penultimate is true; and I say this concerning the motions of the celestial bodies, for the 3500 miles of extra distance they have when in the east rather than overhead neither increase nor diminish them by any perceptible demonstration.

Perspective: proportional diminution with distance

The rule of apparent size loss, applied to why celestial bodies show no change from horizon to overhead

Painting & Perspective Astronomy

Under the heading Perspective, Leonardo states a rule for how objects lose apparent size as they recede: of two things moved back by an equal distance, the one already farther diminishes less, in proportion to their original distances from the eye. He works the case of bodies t and e, whose distances stand in a fivefold ratio, showing the nearer body loses half its size while the farther loses only a fifth. He applies this to the heavens, arguing that the extra 3,500 miles a celestial body lies away at the horizon rather than overhead produces no perceptible change in its apparent size. A diagram of two parallel lines with lettered points accompanies the argument.

On this page

Proportional diminution of equally displaced bodies

Of two things moved back from the eye by an equal distance, the one that was already the farther diminishes less. The proportion of the diminution matches the proportion of their original distances from the eye: doubling the distance of the nearer body t halves its size, while the same displacement shrinks the farther body e by only a fifth.

Celestial bodies show no perceptible size change from horizon to zenith

Leonardo applies the diminution rule to the motions of the celestial bodies. The extra 3,500 miles of distance a body has when in the east rather than directly overhead neither increases nor decreases its apparent size by any perceptible demonstration.

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