Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/87

 But this rectangle, because its breadth AB is upposed diminished in infinitum, becomes les than any given pace. And therefore (by Lem. I.) the figures incribed and circumcribed become ultimately equal one to the other; and much more will the intermediate curvilinear figure be ultimately equal to either.

Q.E.D.

The ame ultimate ratio's are alo ratio's of equality, when the breadths AB, BC, DC, &c., of the parallelograms are unequal, and are all diminished in infinitum.

For uppose AF equal to the greatet breadth, and compleat the parallelogram FAaf. This parallelogram will be greater than the difference of the incrib'd and circumcribed figures; but, becaue its breadth AF is diminished in infinitum, it will become les than any given rectangle. Q.E.D.

Hence the ultimate um of thoe evanecent parallelograms will in all parts coincide with the curvilinear figure.

Much more will the rectilinear figure, comprehended under the chords of the evanecent arcs ab, bc, cd &c. ultimately coincide with the curvilinear figure.

And alo the circumcrib'd rectilinear figure comprehended under the tangents of the ame arcs.

And therefore thee ultimate figures (as to their perimeters acE,) are not rectilinear, but curvilinear limits of rectilinear figures.