number of the parallelograms and parts to be augmented, and their magnitudes diminished in infinitum, those ſums will be in the ultimate ratio of the parallelogram in the one figure to the correſpondent parallelogram in the other; that is, (by the ſupposition) in the ultimate ratio of any part of the one quantity to the correſpondent part of the other.
Lemma V.
In ſimilar figures, all ſorts of homologous ſides, whether curvilinear or rectilinear, are proportional; and the area's are in the duplicate ratio of the homologous ſides.
Lemma VI.
If any arc ACB (Pl.2.Fig.1.) given in poſition is by its chord AB, and in any point A in the middle of the continued curvature is touched by a right line AD, produced both ways; then if the points A and B approach one another and meet, I ſay the angle BAD, contained between the chord and the tangent, will be diminiſhed in infinitum, and ultimately will vanish.
For if that angle does not vaniſh, the arc ACB will contain with the tangent AD an angle equal to a rectilinear angle; and therefore the curvature at the point A will not be continued, which is againſt the ſupposition.
