### 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.*