Among concave mirrors of equal diameter, the one of lesser concavity will unite a greater sum of rays at the point where the rays converge, and consequently will kindle fire with greater speed and power. Let the curve o p stand for the aforesaid mirror; a b is the breadth of the rays raining from the sun onto the mirror; n is placed for the centre of the circle of the sphere from which the mirror has taken its concavity; m is the site where most of the reflected rays converge, and where heat of greater power is created than in any other part of the radiant pyramid o p m. The line, or solar ray, that descends from b to p is the one that rebounds at equal angles along the line p m, as is measured by the curve t r, part of the circumference about the point or centre p. The same is done by all the rays that descend from the sun to a b, always striking and rebounding from the mirror at equal angles, as is shown in the angles v, x. That it is true that all the angles enclosing the angle of the incident and reflected line are equal to one another is clearly proved by the lines that descend from the centre of the sphere from which the mirror takes the quality of its concavity, that is from the point n to o p; these lines all fall upon the mirror at equal angles, as the line K l shows, falling between the angles v, x, which are equal because they are right spherical angles. Then equal parts are taken from them with the line K l and the line m l, as is shown in the curve K d being divided in half at i. Whence, if from the equal angles v, x the equal parts i d l and i h l are removed, the remainder c h v and the remainder x d y will be equal. And thus it is concluded that v, x are equal angles to one another.

a b is that mirror of greater concavity, which I set at equal diameter with the mirror o p below; and you can sensibly recognize how, from 6 to 7 toward b a, no part of that mirror sends its rays to the point of true convergence, that is to i, but all of them fall from that i downward, that is to 2 and to 3. The mirror o p is the fortieth part of the greatest circle found in the sphere from which it is caused, and it would be better as the fiftieth, and better and more useful as the hundredth.

That mirror which is less curved is born from a smaller portion of a circle.

Here you see that the concave mirror of half a spherical body has curved hypotenuses, which within themselves enclose the convergence of the reflected rays; these extend at most as far as the fourth part of the diameter of that sphere, and the more so, the smaller the part one can imagine near that limit. This is shown thus. The larger the angle enclosed between the incident ray and the reflected ray, the shorter the convergence of the reflected rays will be. And conversely, that convergence will be longer, the smaller that angle is. See the angle d m h, which casts the point o born from the base m g, and see the converse of the angle s 3 b, which casts the pyramid at b, born from the base f 3. All the lines that derive from K are those that cut the lower angle in half, that is the angle enclosed between the junction of the incident line and the reflected line.