Let Ba be an arc described in one entire oscillation, C the lowest point of the cycloid, and CZ half the whole cycloidal arc, equal to the length of the pendulum; and let it be required to find the resistance of the body in[Pg 308] any place D. Cut the indefinite right line OQ in the points O, S, P, Q, so that (erecting the perpendiculars OK, ST, PI, QE, and with the centre O, and the asymptotes OK, OQ, describing the hyperbola TIGE cutting the perpendiculars ST, PI, QE in T, I, and E, and through the point I drawing KF, parallel to the asymptote OQ, meeting the asymptote OK in K, and the perpendiculars ST and QE in L and F) the hyperbolic area PIEQ may be to the hyperbolic area PITS as the arc BC, described in the descent of the body, to the arc Ca described in the ascent; and that the area IEF may be to the area ILT as OQ to OS. Then with the perpendicular MN cut off the hyperbolic area PINM, and let that area be to the hyperbolic area PIEQ as the arc CZ to the arc BC described in the descent. And if the perpendicular RG cut off the hyperbolic area PIGR, which shall be to the area PIEQ as any arc CD to the arc BC described in the whole descent, the resistance in any place D will be to the force of gravity as the area to the area PINM. For since the forces arising from gravity with which the body is urged in the places Z, B, D, a, are as the arcs CZ, CB, CD, Ca and those arcs are as the areas PINM, PIEQ, PIGR, PITS; let those areas be the exponents both of the arcs and of the forces respectively. Let Dd be a very small space described by the body in its descent: and let it be expressed by the very small area RGgr comprehended between the parallels RG, rg; and produce rg to h, so that GHhg and RGgr may be the contemporaneous decrements of the areas IGH, PIGR. And the increment , or , of the area will be to the decrement RGgr, or Rr × RG, of the area PIGR, as to RG; and therefore as to OR × GR or OP × PI, that is (because of the equal quantities OR × HG, OR × HR - OR × GR, ORHK - OPIK, PIHR and PIGR + IGH), as to OPIK. Therefore if the area be called Y, and RGgr the decrement of the area PIGR be given, the increment of the area Y will be as PIGR - Y.