Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/98

 always be the ame as before and therefore the ame with the ratio of $$\scriptstyle{AB^2}$$ to $$\scriptstyle{Ab^2}$$. Q.E.D.

And if we uppose 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; nevertheles the angles D, d, being determined by the ame law, will always draw nearer to each other, and approach nearer to each other than any aigned difference, and therefore by Lem. 1, will at lat be equal; and therefore the lines BD, bd arc in the ame ratio to each other as before.

Therefore ince the tangents AD, Ad, the arcs AB, Ab, and their ines, BC, bc, become ultimately equal to the chords AB, Ab, their quares will ultimately become as the ubtenes BD, bd.

Their quares are alo ultimately as the vered ines of the arcs, biecting the chords, and converging to a given point. For thoe vered ines are as the ubtenes BD, bd.

And therefore the vered ine is in the duplicate ratio of the time in which a body will describe the arc with a given velocity.

The rectilinear triangles ADB, Adb are ultimately in the triplicate ratio of the ides AD, Ad, and in a equiplicate ratio of the ides DB, db; as being in the ratio compounded of the ides AD to DB, and of Ad to db. So alo the triangles ABC, Abc are ultimately in the triplicate ratio of the ides BC, bc. What I call the equiplate ratio is the ubduplicate of the triplicate, as being compounded of the imple and ubduplicate ratio.

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