to that rectangle. Then A will be the place from whence the other body fell. For compleating the rectangle DRSE, ſince the area AbFD is to the area DFGE as VV to 2VI, and therefore as , that is, as half the whoſe velocity to the increment of the velocity of the body falling by the unequable force; and in like manner the area PQRD to the area DRSE, as half the whole velocity to the increment of the velocity of the body falling by the uniform force; and ſince thoſe increments (by reaſon of the equality of the naſcent times) are as the generating forces, that is, as the ordinates DF, DR, and conſequently as the naſcent area's DFGE, DRSE; therefore ex æquo the whole areas ABFD, PQRD will be to one another as the halves of the whole velocities, and therefore, becauſe the velocities are equal, they become equal alſo.
Cor. 2. Whence if any body be projected either upwards or downwards with a given velocity from any place D, and there be given the law of centripetal force acting on it, its velocity will be found in any other place as e, by erecting the ordinate eg, and taking that velocity to the velocity in the place D, as a right line whoſe power is the rectangle PQRD, either increaſed by the curvilinear area Dfge, if the place e is below the place D, or diminiſhed by the ſame area DFge if it be higher, is to the right line whole power is the rectangle PQRD alone.
Cor. 3. The time is alſo known by erecting the ordinate em reciprocally proportional to the ſquare root of PQRD + or - DFge, and taking the time in which the body has deſcribed the line De, to the time in which another body has fallen with an
