always be the ſame as before and therefore the ſame with the ratio of to . Q.E.D.
Case 3. And if we ſuppose the angle D not to be given, but that right line BD converges to a given point, or is determined by any other condition whatever; nevertheleſs the angles D, d, being determined by the ſame law, will always draw nearer to each other, and approach nearer to each other than any aſſigned difference, and therefore by Lem. 1, will at laſt be equal; and therefore the lines BD, bd arc in the ſame ratio to each other as before.
Cor 1. Therefore ſince the tangents AD, Ad, the arcs AB, Ab, and their ſines, BC, bc, become ultimately equal to the chords AB, Ab, their ſquares will ultimately become as the ſubtenſes BD, bd.
Cor 2. Their ſquares are alſo ultimately as the verſed ſines of the arcs, biſecting the chords, and converging to a given point. For thoſe verſed ſines are as the ſubtenſes BD, bd.
Cor 3. And therefore the verſed ſine is in the duplicate ratio of the time in which a body will describe the arc with a given velocity.
Cor 4. The rectilinear triangles ADB, Adb are ultimately in the triplicate ratio of the ſides AD, Ad, and in a ſeſquiplicate ratio of the ſides DB, db; as being in the ratio compounded of the ſides AD to DB, and of Ad to db. So alſo the triangles ABC, Abc are ultimately in the triplicate ratio of the ſides BC, bc. What I call the ſeſquiplate ratio is the ſubduplicate of the triplicate, as being compounded of the ſimple and ſubduplicate ratio.
Cor 5. And because DB, db are ultimately parallel and in the duplicate ratio of the lines AD, Ad, the ultimate curvilinear areas ADB, Adb will be
