Isaac Newton

“ Supposing that the centripetal force is reciprocally proportional to the square of the distance of the places from the centre; it is required to define the spaces which a body, falling directly, describes in given times. ”

“ For (by cor. 7. prop. 16.) the velocity of a body describing a parabola RPB about the centre S, in any place P, is equal to the velocity of a body uniformly describing a circle about the same centre S at half the interval SP. Let the breadth CP of the parabola be diminished in infinitum, so as the parabolic arc PfB may come to coincide with the right line CB, the centre S with the vertex B, and the interval SP with the interval BC, and the proposition will be manifest. ”

“ 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. ”