Lemma VII.
The same things being ſupposed, I ſay that the ultimate ratio of the arc, chord, and tangent, any one to any other, is the ratio of equality. Pl. 2. Fig. 1.
For while the point B approaches the point A, conſider always AB and AD as produc'd to the remote point b and d; and parallel to the ſecant BD draw bd; and let the arc Acb be always ſimilar to the arc ACB. Then, ſupposing the points A and B to coincide, the angle dAb will vaniſh, by the preceding lemma; and therefore the right lines Ab, Ad (which are always finite), and the intermediate arc Acb, will coincide, and become equal among themſelves. Wherefore, the right lines AB, AD, and the intermediate arc ACB (which are always proportional to the former) will vaniſh; and ultimately acquire the ratio of equality. Q.E.D.
COR 1. Whence if through B we draw BF parallel to the tangent, always cutting any right line AF passing through A in F, this line BF will ultimately be in the ratio of equality with the evaneſcent arc ACB; becauſe compleating the parallelogram AFBD, it is always in a ratio of equality with AD.
COR 2. If through B and A more right lines are drawn, as BE, BD, AF, AG, cutting the tangent AD and its parallel BF; the ultimate ratio of all abſciſſas AD, AE, BF, BG and of the chord and arc AB, any one to any other, will be the ratio of equality.
