A way of finding the angle of incidence upon a spherical mirror.

Upon the convex mirror let two points be given; and let one be the eye and the other the object. It is asked for the position of the image of that object, that is, the angle common to the incident ray and the reflected ray. First you will make the semicircle L c m, which represents the convex, or spherical, mirror. Then place the two points a, b at a distance of your own choosing. And from those points draw two straight lines that meet at the centre d, which will be the lines a d and b d. Then from the two points e, f, where the meeting of the two lines cuts the circumference of the circle, draw the straight line e f, extended indefinitely to g h. Then draw the straight line i K, which is parallel to the line g h and tangent to the circle at c. And this will be our guide. And from the two given points a, b draw the perpendiculars onto the line i K, so that they cut it and pass in a straight, indefinite line below to o and n. Having our figuration in being, it is necessary to define its members. For the two triangles made upon the line i K can never be similar unless they are equal to one another; and, not being able to be similar, it is necessary to make one similar to it so that their proportions may be demonstrated. This will be brought about by continuing the straight line b c to c n, so that it cuts the line a n; and so the triangle i c n will be made, which will have its sides in the same proportion as the sides of the greater triangle, as the base i c has with the base c K. Then, to make the second triangle proportioned to the triangle a i c, you will continue the straight line a c as far as c o, so that it cuts the line b o; and so you will have made the fourth triangle proportioned to the third a i c, as the second to the first; and at the meeting of their four angles in c will be the angle of incidence that we were seeking.