LEM

LEM. II. If in any figure $AaE$ (Pl. CCIV. fig. 2.) terminated by the right line $Aa$, $AE$, and the curve $aE$, there be inscribed any number of parallelograms $Ab$, $Bc$, $Cd$, &c. 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 completed. Then if the breadth of those parallelograms be supposed to be diminished, and their number augmented in infinitum; the ultimate ratios which the inscribed figure $AkbLcMdD$, the circumscribed figure $AalbmendoE$, and curvilinear figure $AabedE$, will have to one another, 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, (from the equality of all their bases), the rectangle under one of their bases $Kb$, and the sum of their altitudes $Aa$, that is, the rectangle $AB/a$. But this rectangle, because its breadth $AB$ is supposed diminished in infinitum, becomes less than any given space. And therefore, by lem. 1. the figures inscribed and circumscribed become ultimately equal the one to the other; and much more will the intermediate curvilinear figure be ultimately equal to either.