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Library/Codex Arundel/f. 20r (transmitted light)

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Codex Arundel — f. 20r (transmitted light)

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British Library, London — Arundel MS 263

Dell'angolo della incidenzia.

Figura con lettere e numeri.

Li stremi delle proporzioni partiti in parte equali gieneran proporzioni equali alla prima.

Dati due punti in vari siti sopra un cerchio, che l'uno rapresenti un luminoso e l'altro l'ochio, dimandasi in che punto d'esso cerchio sarà più alluminato. Sia a, b li due dati puncti; p y q è il dato cerchio. Dal punto a e dal punto b si tiri due linie, che concorrino in centro t. Poi dal punto a tira la retta a c che tochi la circunferenzia del cerchio p y q; e 'l simile fa dal b in d, e così arai fatto l'angolo a n b. Il quale sarebbe l'angolo della incidenzia che noi cerchiamo, se esso tocassi il cerchio; ma, per essere lui troppo alto, e' ci bisogna trovare un modo d'abassarlo, e che l'alteza de' sua lati restino nella medesima proporzione di prima. La qual cosa farem colla regola data di sopra. Noi discendereno per la linia a g i tanto presso al cerchio quanto è quasi la saetta y 9 dell'arco 3 y 7; e dal lato opposito fareno il simile nella basseza della linia b 7; ma sarà parte che sarà nella medesima proporzione colla parte del a g, qual'è la proporzione che ha tutto b 7 con tutto a g. E di queste parti dette tirerai dalli lor superiori stremi la linia g v e la linia o r, e che l'una e l'altra tochi il cerchio. E così nella intersegazione d'esse due linie fia l'angolo della incidenzia nella linia y t che viene dal centro del detto cerchio.

Of the angle of incidence.

Figure with letters and numbers.

The extremes of proportions divided into equal parts will generate proportions equal to the first. Given two points in various sites upon a circle, of which the one represents a luminous body and the other the eye, it is asked at what point of this circle it will be most illuminated. Let a, b be the two given points; p y q is the given circle. From point a and from point b draw two lines that meet at the center t. Then from point a draw the straight line a c that touches the circumference of the circle p y q; and do the like from b to d, and thus you will have made the angle a n b. Which would be the angle of incidence that we seek, if it touched the circle; but, it being too high, we need to find a way to lower it, so that the height of its sides remain in the same proportion as before. Which thing we shall do with the rule given above. We shall descend by the line a g i as close to the circle as is almost the sagitta y 9 of the arc 3 y 7; and on the opposite side we shall do the like in the lowering of the line b 7; but it will be a part that will be in the same proportion with the part of a g as is the proportion that the whole b 7 has with the whole a g. And from these said parts you will draw, from their upper extremes, the line g v and the line o r, and let each of them touch the circle. And thus in the intersection of these two lines will be the angle of incidence on the line y t that comes from the center of the said circle.

Finding the Angle of Incidence on a Circle

The most illuminated point between a light and the eye (f. 20r)

Optics & Light Geometry

Under the heading 'Of the angle of incidence' Leonardo poses two points on a circle, one a luminous body and one the eye, and asks at what point of the circle it is most illuminated. He constructs tangents a c and b d to form the angle a n b, then, finding it too high, lowers the construction along a g i and b 7 while keeping the sides in the same proportion, so that lines g v and o r meet on the radius y t at the angle of incidence. A construction of crossing lines fills the top of the leaf.

On this page

The most illuminated point on a circle and its angle of incidence

Given two points a and b on the circle p y q, one a light and the other the eye, Leonardo seeks the most illuminated point. Drawing lines from a and b that meet at the center t, and the tangents a c and b d, he forms the angle a n b, which would be the sought angle of incidence if it touched the circle.

Lowering the construction while keeping proportion

Because the angle a n b sits too high, he lowers it by descending along a g i near the circle by about the sagitta y 9 of the arc 3 y 7, and doing the like on b 7 in the same proportion. Lines g v and o r drawn from the upper extremes then touch the circle, and their intersection gives the angle of incidence on the radius y t.

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