Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/99

 1. (by the nature of parabola) two thirds of the rectilinear triangles ADB, Adb and the egments AB, Ab will be one third of the ame triangles. And thence thoe areas and those egments will be in the triplicate ratio as well of the tangents AD, Ad, as of the chords and arcs AB, Ab.

But we have all along upposed the angle of contact to be neither infinitely greater nor infinitely les, than the angles of contact made by circles and their tangents; that is, that the curvature at the point A is neither infinitely mall nor infinitely great, or that the interval AJ is of a finite magnitude. For DB may be taken as AD3: in which cae no circle can be drawn through the point A, between the tangent AD and the curve AB, and therefore the angle of contact will be infinitely les than those of circles. And by a like reaoning, if DB be made uccessfully as AD4, AD5, AD6, AD7, &c., we hall have a eries of angles of contact, proceeding in infinitum, wherein every ucceeding term is infinitely les than the preceding. And if DB be made successively as AD2, AD3/2, AD4/3, AD5/4, AD6/5, AD7/6, &c., we shall have another infinite eries of angles of contact, the first of which is of the ame ort with those of circles, the econd infinitely greater, and every ucceeding one infinitely greater than the preceding. But between any two of these angles another eries of intermediate angles of contact may be interpoed, proceeding both ways in infinitum, wherein every ucceeding angle hall be infinitely greater, or infinitely les than the preceding. As if between the Rh