Angle of incidence on a spherical mirror
Geometric construction locating where a ray reflects on a concave sphere
On the left half of the sheet (folio 23v) Leonardo works out where the angle of incidence falls when an eye at one point sees an object at another reflected in a spherical mirror. He argues that the incidence angle can never lie outside the cathetus running from eye to object through the centre of the sphere, and gives a step-by-step compass construction on the circle m p y to locate the reflection point n. A geometric proof follows, using the equal angles m r n and p t n and the properties of the circle m r t p. The right-hand page (folio 22r) is blank apart from the folio number.
On this page
The incidence angle is bounded by the eye-to-object cathetus
The angle of incidence will never fall outside the cathetus that arises between the eye and the object and ends at the centre of the spherical mirror. Thus if the space between the eye and the object were a b, its cathetus in the spherical mirror K o s would be a b c.
Compass construction of the reflection point n
Let a and b be the two given points on the circle m p y, and draw a c and b c to its centre. After building the base e b of triangle e b c and swinging the arc f h, where line a b cuts the arc at g a line g c is drawn to the centre; where it cuts the circle at o the space o p is carried to the opposite side onto m n, and at n lies the point where the angle of incidence is caused.
Proof by the equal angles m r n and p t n
It is proved by the equal angles m r n and p t n, which enclose the angle of incidence a n b; these are shown equal by the definition of the circle m r t p, in which the sides r n and t n are equal and the curves m r and t p are equal, being on the parallels u K and s x with equal curvatures.
Triangle-and-circle diagram
The upper-left figure shows a triangle set over a circle with the centre marked, carrying the letter-labels named in the surrounding notes. Further ink labels on the diagram were not included in the transcription.