Cor. 3. And therefore in all our reaſoning about ultimate ratio's, we may freely uſe any one of thoſe lines for any other.
Lemma VIII.
If the right lines AR, BR, with the arc ACB, the chord AB, and the tangent AD, conſtitute three triangles RAB, RACB, RAD; and the points A and B approach and meet: I ſay that the ultimate form of these evaneſcent triangles is that of ſimilitude, and their ultimate ratio that of equality.
For while the point B approaches towards the point A, consider always AB, AD, AR, as produced to the remote points b, d, and r, and rbd drawn parallel to RD, and let the arc Acb be always ſimilar to the arc ACB. Then ſupposing the points A and B to coincide the angle bAd will vaniſh; and therefore the three triangles rAb, rAcb, rAd will coincide, and on that account become both ſimilar and equal. And therefore the triangles RAB, RACB, RAD which are always ſimilar and proportional to these, will ultimately become both ſimilar and equal among themselves. Q.E.D.
Cor. And hence in all our reaſonings about ultimate ratio's, we may indifferently uſe any one of thoſe triangles for any other.
