But this rectangle, because its breadth AB is ſupposed diminished in infinitum, becomes leſs than any given ſpace. And therefore (by Lem. I.) the figures inſcribed and circumſcribed become ultimately equal one to the other; and much more will the intermediate curvilinear figure be ultimately equal to either.
Q.E.D.
Lemma III.
The ſame ultimate ratio's are alſo ratio's 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 compleat the parallelogram FAaf. This parallelogram will be greater than the difference of the inſcrib'd 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 circumſcrib'd 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.
