Lemma IV.

*If in two figures* AacE, PprT, (Pl.i.Fig.7.) *you inſcribe (as before) two ranks of parallelograms, an equal number in each rank, and when their breadths are diminiſhed *in infinitum*, the ultimate ratio's of the parallelograms in one figure to thoſe in the other each to each reſpectively, are the ſame; I ſay that thoſe two figures *AacE, PprT*, are to one another in that ſame ratio.*

For as the parallelograms in the one are ſeverally to the parallelograms in the other, ſo (by compoſition) is the ſum of all in the one to the ſum of all in the other; and ſo is the one figure to the other, becauſe (by Lem. 3.) the former figure to the former ſum, and the latter figure to the latter sum are both in the ratio of equality. *Q. E. D.*

COR. Hence if two quantities of any kind are any how divided into an equal number of parts: and thoſe parts, when their number is augmented and their number diminished *in infinitum*, have a given ratio one to the other, the firſt to the firſt, the ſecond to the ſecond, and ſo on in order; the whole quantities will be one to the other in that ſame given ratio. For if, in the figures of this lemma, the parallelograms are taken one to the other in ratio of the parts, the ſum of the parts will always be as the ſum of the parallelograms; and therefore ſuppoſing the