If in some figure AacE, with the right lines Aa, AE and the curve acE in
place, some number of parallelograms are inscribed Ab, Bc, Cd, &c. with equal
bases AB, BC, CD, &c. below, and with the sides Bb, Cc, Dd, &c. maintained
parallel to the side of the figure Aa; & with the parallelograms aKbl, bLcm,
cMdn, &c. filled in. Then the width of these parallelograms may be diminished
and the number may be increased to infinity : I say that the final ratios which
the inscribed figure AKbLcMdD, the circumscribed figure AalbmcndoE, and to the
curvilinear figure AabcdE have in turn between each other, are ratios of
equality.
For the difference of the inscribed and circumscribed figures is the sum of the
parallelograms Kl, Lm, Mn, Do, that is (on account of the equal bases) the
rectangle under only one of the bases Kb and the sum of the heights Aa, that is,
the rectangle ABla. But this rectangle, because with the width of this AB
diminished indefinitely, becomes less than any given [rectangle] you please.
Therefore (by lemma I) both the inscribed and circumscribed figures finally
become equal, and much more [importantly] to the intermediate curvilinear
figure.
Q. E. D.
Si in figura quavis AacE, rectis Aa, AE & curva acE comprehensa, inscribantur parallelogramma quotcunque Ab, Bc, Cd, &c. sub basibus AB, BC, CD, &c. aequalibus, & lateribus Bb, Cc, Dd, &c. figurae lateri Aa parallelis contenta; & compleantur parallelogramma aKbl, bLcm, cMdn, &c. Dein horum parallelogrammorum latitudo minuatur, & numerus augeatur in infinitum: dico quod ultimae rationes quas habent ad se invicem figura inscripta AKbLcMdD, circumscriptae AalbmcndoE, & curvilinea AabcdE, sunt rationes aequalitatis.
Nam figurae inscripae & circumscriptae differentia est summa parallelogrammorum
Kl, Lm, Mn, Do, hoc est (ob aequales omnium bases) rectangulum sub unius basi Kb
& altitudinum summa Aa, id est, rectangulum ABla. Sed hoc rectangulum, eo quod
latitudo eius AB in infinitum minuitur, fit minus quovis dato. Ergo (per lemma
I) figura inscripta & circumscripta & multo magis figura curvilinea intermedia
fiunt ultimo aequales. Q. E. D.