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Codex Arundel — f. 20r

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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.

On the angle of incidence.

Figure with letters and numbers.

The extremes of the proportions, divided into equal parts, will generate proportions equal to the first. Given two points in various positions upon a circle, one representing a luminous body and the other the eye, it is asked at what point of that 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 centre t. Then from point a draw the straight line a c so that it touches the circumference of the circle p y q; and do the like from b to d, and so you will have made the angle a n b. This would be the angle of incidence that we seek, if it touched the circle; but, being too high, we must find a way to lower it so that the height of its sides remain in the same proportion as before. This we shall do by the rule given above. We shall descend along the line a g i as near to the circle as is roughly 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 shall be a part that is in the same proportion to the part of a g as is the proportion that all of b 7 has to all of a g. And from the upper ends of these said parts you shall draw the line g v and the line o r, so that each of them touches the circle. And so, at the intersection of these two lines, the angle of incidence will lie on the line y t that comes from the centre of the said circle.

On the angle of incidence: the best-lit point of a circle

A construction to find where a circle is most illuminated by a light and seen by the eye

Optics & Light Geometry

Under the heading 'On the angle of incidence', Leonardo sets a problem: given two points on a circle, one a light and the other the eye, at what point of the circle is it most illuminated? The lettered and numbered figure carries the construction, drawing lines from a and b to the centre t, tangents a c and b d to form the angle a n b, then lowering that angle by proportional parts along a g and b 7 until two tangent lines g v and o r meet on the central line y t. A star of ruled lines with two arcs at the foot of the page realises the construction.

On this page

The most illuminated point of a circle

Given a luminous point and the eye set at two places on a circle, the problem is to find at what point the circle is most illuminated. The sought angle of incidence is shown to lie on the line y t drawn from the centre of the circle.

Constructing and lowering the angle a n b

Lines from points a and b meet at the centre t, and tangents a c and b d form the angle a n b. Being too high, this angle is lowered by proportional parts measured along a g and b 7, so that two new tangent lines g v and o r meet at the true angle of incidence.

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