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

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Sect. 1.
of Natural Philosophy.
53

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

Scholium

But we have all along ſupposed the angle of contact to be neither infinitely greater nor infinitely leſs, than the angles of contact made by circles and their tangents; that is, that the curvature at the point A is neither infinitely ſmall nor infinitely great, or that the interval AJ is of a finite magnitude. For DB may be taken as AD3: in which caſe 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 leſs than those of circles. And by a like reaſoning, if DB be made ſuccessfully as AD4, AD5, AD6, AD7, &c., we ſhall have a ſeries of angles of contact, proceeding in infinitum, wherein every ſucceeding term is infinitely leſs 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 ſeries of angles of contact, the first of which is of the ſame ſort with those of circles, the ſecond infinitely greater, and every ſucceeding one infinitely greater than the preceding. But between any two of these angles another ſeries of intermediate angles of contact may be interpoſed, proceeding both ways in infinitum, wherein every ſucceeding angle ſhall be infinitely greater, or infinitely leſs than the preceding. As if between the

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terms