Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/91

 number of the parallelograms and parts to be augmented, and their magnitudes diminished in infinitum, those ums will be in the ultimate ratio of the parallelogram in the one figure to the correpondent parallelogram in the other; that is, (by the upposition) in the ultimate ratio of any part of the one quantity to the correpondent part of the other.

In imilar figures, all orts of homologous ides, whether curvilinear or rectilinear, are proportional; and the area's are in the duplicate ratio of the homologous ides.

If any arc ACB (Pl.2.Fig.1.) given in poition 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 ay the angle BAD, contained between the chord and the tangent, will be diminihed in infinitum, and ultimately will vanish.

For if that angle does not vanih, 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 againt the upposition.