b - a - c

So great a multitude of intersections is made that touch the circles, because in one of these intersections is the angle of contingency, but it is not known which. Now let us imagine infinite circles one within another, so that we know the last, smallest one would end in a point. And thus we shall imagine infinite intersections one within another, and from one to another draw a line, which would be curved.

p - m a o - b - r - c - b - t S n

Make the small circle within the given circle and draw the line p S tangent to the given circle, and likewise the line o m. Then do the same for the smaller circle; the third circle is the point. Then draw two straight lines from the intersections, that is a b and b c, to a point. Then draw two perpendiculars upon the lines made, which will be n b and n r, and set the foot of the compass at n and make the circle that passes over the two intersections and the centre of the circle; and where such a curve cuts the given circle, there is the angle of contingency.

But the point n might sometimes be at a very great distance, and you could not make such an angle. Then you will take a thin, straight iron wire and bend it so that it touches the two intersections and the centre of the circle; and where that iron wire touches the periphery of the given circle, there will be the angle of contingency.

Given the eye and the whole length of the reflected line, the length of the incident line is asked. It cannot be given according to the intention of the poser, because it may be of infinite shortness and length.

Of the intersection, or lines that intersect at S, no account is taken in the proportions, but they serve only to make the lines b d and a c, which must be in contingency with the given circle K g L.

The triangles a i g and b g f are proportional in all their three sides through being placed upon the straight line i f (a figure follows). Here all the proportions born of the first proportion answer to one another: and this is that b, the given point, is placed twice as far from the angle of contingency g as a is; whence it follows that b d is double a c, and likewise b f to a i, and so f d to i c; and this arises because the pyramid b e d, being cut in the middle of its length at a c, by the ninth [proposition] of this [book] it follows that all the lines that can be drawn from the base b d to the apex e will cut the line a c in the same proportion as the base is cut.