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
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.