The same ultimate ratios are also ratios of equality, when the breadths, AB, BC, DC, &c., of the parallelograms are unequal, and are all diminished in infinitum.
For suppose AF equal to the greatest breadth, and complete the parallelogram FAaf. This parallelogram will be greater than the difference of the inscribed and circumscribed figures; but, because its breadth AF is diminished in infinitum, it will become less than any given rectangle. Q.E.D.
COR. 1. Hence the ultimate sum of those evanescent parallelograms will in all parts coincide with the curvilinear figure.
COR. 2. Much more will the rectilinear figure comprehended under the chords of the evanescent arcs ab, bc, CD, &c., ultimately coincide with the curvilinear figure.
COR. 3. And also the circumscribed rectilinear figure comprehended under the tangents of the same arcs.
COR. 4. And therefore these ultimate figures (as to their perimeters acE) are not rectilinear, but curvilinear limits of rectilinear figures.