Page:Newton's Principia (1846).djvu/314

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 aysmptotes 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 $\scriptstyle {\frac {OR}{OQ}}$ IEF - IGH 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 GHhg - $\scriptstyle {\frac {Rr}{OQ}}$ IEF, or Rr $\scriptstyle \times$ HG - $\scriptstyle {\frac {Rr}{OQ}}$ IEF, of the area $\scriptstyle {\frac {OR}{OQ}}$ IEF - IGH will be to the decrement RGgr, or Rr $\scriptstyle \times$ RG, of the area PIGR, as HG - $\scriptstyle {\frac {IEF}{OQ}}$ to RG; and therefore as OR $\scriptstyle \times$ HG - $\scriptstyle {\frac {OR}{OQ}}$ IEF to OR $\scriptstyle \times$ GR or OP $\scriptstyle \times$ PI, that is (because of the equal quantities OR $\scriptstyle \times$ HG, OR $\scriptstyle \times$ HR - OR $\scriptstyle \times$ GR, ORHK - OPIK, PIHR and PIGR + IGH), as PIGR + IGH - $\scriptstyle {\frac {OR}{OQ}}$ IEF to OPIK. Therefore if the area $\scriptstyle {\frac {OR}{OQ}}$ IEF - IGH be called Y, and RGgr the decrement of the area PIGR be given, the increment of the area Y will be as PIGR - Y.

Then if V represent the force arising from the gravity, proportional to the arc CD to be described, by which the body is acted upon in D, and R be put for the resistance, V - R will be the whole force with which the body is urged in D. Therefore the increment of the velocity is as V - R and the particle of time in which it is generated conjunctly. But the velocity itself is as the contemporaneous increment of the space described di-