MAT 320 Introduction to Analysis Fall 1996

Newton's Left-Sum Right-Sum Lemma

Translation If in an arbitrary figure AacE bounded by the lines Aa, AE and the curve AcE,are inscribed a number of rectangles; Ab, Bc, Cd, etc. on equal bases AB, BC, CD, etc., and with sides Bb, Cc, Dd, etc. parallel to the side Aa of the figure; let us construct the parallelograms aKBl, bLcm, cMdn, etc. Then let the width of these parallelograms be diminished, and their number increased to infinity: I say that the ultimate ratios which exist between the inscribed figure AKbLcMdD, the circumscribed AalbmcndoE and the curvilinear AabcdE are ratios of equality.
In fact the difference between the inscribed and circumscribed figures is the sum of the parallelograms Kl + Lm + Mn + Do, which is (since their bases are all equal) the rectangle on a single base Kb and with height the sum Aa, i.e. the rectangle ABla. But this rectangle, since its width AB is being diminished infinitely, will become smaller than any given quantity. Thus, by Lemma I, the inscribed and circumscribed figures, and certainly all the more the intermediate curvilinear figure, are ultimately equal. Q.E.D.