Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/88

 If in two figures AacE, PprT, (Pl..Fig.7.) you incribe (as before) two ranks of parallelograms, an equal number in each rank, and when their breadths are diminihed in infinitum, the ultimate ratio's of the parallelograms in one figure to thoe in the other each to each repectively, are the ame; I ay that thoe two figures AacE, PprT, are to one another in that ame ratio.



For as the parallelograms in the one are everally to the parallelograms in the other, o (by compoition) is the um of all in the one to the um of all in the other; and o is the one figure to the other, becaue (by Lem. 3.) the former figure to the former um, 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 thoe parts, when their number is augmented and their number diminished in infinitum, have a given ratio one to the other, the firt to the firt, the econd to the econd, and o on in order; the whole quantities will be one to the other in that ame given ratio. For if, in the figures of this lemma, the parallelograms are taken one to the other in ratio of the parts, the um of the parts will always be as the um of the parallelograms; and therefore uppoing the