Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/86

 other than by any given difference, become ultimately equal.

If you deny it; uppose them to be ultimately unequal, and let D be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D; which is againt the uppoition.

If in any figure AacE (Pl.1.Fig.6.) terminated by the right lines Aa, AE, and the curve acE, there be incrib'd any number of parallelograms Ab, Bc, Cd,'' &c. comprehended under equal baes AB, BC, CD, &c. and the ides Bb, Cc, Dd, &c. parallel to one ide Aa of the figure; and the parallelograms aKbl, bLcm, cMdn, &c. are compleated. Then if the breadth of thoe parallelograms be uppo'd to be diminihed, and their number to be augmented in infinitum: I ay that the ultimate ratio's which the incrib'd figure AKbLcMdD, the circumcribed figure AalbmcndoE, and curvilinear figure AabcdE, will have to one another, are ratio's of equality.''

For the difference of the incrib'd and circumcrib'd figures is the sum of the parallelograms Kl, Lm, Mn, Do, that is, (from the equality of all their baes) the rectangle under one of their baes Kb and the um of their altitudes Aa, that is, the rectangle ABla,