Secondly, and consequently: if two unequal arches of the same circle be made by the bowing of two strait lines equal to them, the flexion of the longer line, whilst it is bowed into the greater arch, is greater than the flexion of the shorter line, whilst it is bowed into the lesser arch, according to the proportion of the arches themselves; and consequently, the crookedness of the greater arch is to the crookedness of the lesser arch, as the greater arch is to the lesser arch. Thirdly: if two unequal circles and a strait line touch one another in the same point, the crookedness of any arch taken in the lesser circle, will be greater than the crookedness of an arch equal to it taken in the greater circle, in reciprocal proportion to that of the radii with which the circles are described; or, which is all one, any strait line being drawn from the point of contact till it cut both the circumferences, as the part of that strait line cut off by the circumference of the greater circle to that part which is cut off by the circumference of the lesser circle. For let A B and A C (in the second figure) be two circles, touching one another, and the strait line A D in the point A; and let their centres be E and F; and let it be supposed, that as A E is to A F, so is the arch A B to the arch A H. I say the crookedness of the arch A C is to the crookedness of the arch A H, as A E is to A F. For let the strait line A D be supposed to be equal to the arch A B, and the strait line A G to the arch A C; and let A D, for example, be double to A G. Therefore, by reason of the likeness of the arches A B and A C, the strait line A B will be double to the strait line A C, and the radius A E double to the radius A F, and the arch A B double to the arch A H. And because the strait line A D is so bowed to be coincident with the arch A B equal to it, as the strait line A G is bowed to be coincident with the arch A C equal also to it, the flexion of the strait line A G into the crooked line A C will be equal to the flexion of the strait line A D into the crooked line A B. But the flexion of the strait line A D into the crooked line A B is double to the flexion of the strait line A G into the crooked line A H; and therefore the flexion of the strait line A G into the crooked line A C is double to the flexion of the same strait line A G into the crooked line A H. Wherefore, as the arch A B is to the arch A C or A H; or as the radius A E is to the radius A F; or as the chord A B is to the chord A C; so reciprocally is the flexion or uniform crookedness of the arch A C, to the flexion or uniform crookedness of the arch A H, namely, here double. And this may by the same method be demonstrated in circles whose perimeters are to one another triple, quadruple, or in whatsoever given proportion. The crookedness therefore of two equal arches taken in several circles are in proportion reciprocal to that of their radii, or like arches, or like chords; which was to be demonstrated.