OF NATURAL PHILOSOPHY 95 BOOK I. OF THE MOTION OF BODIES. SECTION I. Of the method of first and last ratios of quantities, by the help wJicreoj we demonstrate the propositions that follow. LEMMA I. Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal. If you deny it, suppose 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 against the supposition, LEMMA II. If in any figure AacE, terminated by the right (f lines A a. AE, and the curve acE, there be in scribed any number of parallelograms Ab, Be, Cd, fyc., comprehended under equal bases AB, BC, CD, ^«c., and the sides, Bb, Cc, Dd, ^c., parallel to one side Aa of the figure ; and the parallelograms aKbl, bLcm, cMdn, §*c., are com pleted. Then if the breadth of those parallelo- \ grams be supposed to be diminisJied, and their X BF C D