Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/281

 wheel. Imagine this wheel to proceed in the great circle ABL from A through B towards L. and in its progres to revolve in uch a manner that the arcs AB, PB may be always equal the one to the other, and the given point P in the perimeter of the wheel may decribe in the mean time the curvilinear path AP. Let AP be the whole curvilinear path decribed ince the wheel touched the globe in A, and the length of this path AP will be to twice the vered line of the arc $$\scriptstyle \frac 12$$ PB, as 2 CE to CB. For let the right line CE (produced if need be) meet the wheel in V, and join CP, BP, EP, VP; produce CP, and let fall thereon the perpendicular VF. Let PH, VH, meeting in H, touch the circle in P and K and let PH cut VF in G, and to VP let fall the perpendiculars GI, HK. From the centre C with any interval let there be decribed the circle nom, cutting the right line CP in n, the perimeter of the wheel BP in o, and the curvilinear path AP in m; and from the centre V with the interval Va let there be decribed a circle cutting VP produced in q.

Becaue the wheel in its progres always revolves about the point of contact B, it is manifet that the right line BP is perpendicular to that curve line AP which the point P of the wheel decribes, and therefore that the right line VP will touch this curve in the point P. Let the radius of the circle nom be gradually increaed or diminihed o that at lat it become equal to the ditance CP; and by reaon of the imilitude of the evanecent figure Pnomq, and the figure PFGVI, the ultimate ratio of the evanecent lineolæ Pm, Pn, Po, Pq, that is, the ratio of the momentary mutations