Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/93

. And therefore in all our reaoning about ultimate ratio's, we may freely ue any one of thoe lines for any other.

If the right lines AR, BR, with the arc ACB, the chord AB, and the tangent AD, contitute three triangles RAB, RACB, RAD; and the points A and B approach and meet: I ay that the ultimate form of these evanecent triangles is that of imilitude, 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 imilar to the arc ACB. Then upposing the points A and B to coincide the angle bAd will vanih; and therefore the three triangles rAb, rAcb, rAd will coincide, and on that account become both imilar and equal. And therefore the triangles RAB, RACB, RAD which are always imilar and proportional to these, will ultimately become both imilar and equal among themselves. Q.E.D.

. And hence in all our reaonings about ultimate ratio's, we may indifferently ue any one of thoe triangles for any other.