Isaac Newton

“ And therefore the force is as the disŧance of the body from the centre of the ellipsis; and vice versa if the force is as the distance, the body will move in an ellipsis whose centre coincides with the centre of force, or perhaps in a circle into which the ellipsis may degenerate. ”

“ In a space void of resisŧance, if a body revolves in any orbit about an immoveable centre, and in the least time describes any arc just then nascent; and the versed sine of that arc is suppofed to be drawn, bisecting the chord, and produced passing through the centre of force: the centripetal force in the middle of the arc, will be as the versed sine directly and the square of the time inversely. ”

“ If the periodic times are in the sesquiplicate ratio of the radii, and therefore the velocities reciprocally in the subduplicate ratio of the radii; the centripetal forces will be in the duplicate ratio of the radii inversely: and the contrary.
Cor. 7. And universally, if the periodic time is as any power of the radius R, and therefore the velocity reciprocally as the power of the radius; the centripetal force will be reciprocally as the power of the radius: and the contrary. ”

“ And even in resisting mediums, if the description of the areas is accelerated, the directions of the forces deviate from the point in which the radii meet, towards the parts to which the motion tends.
Scholium
A body may be urged by a centripetal force compounded or several forces. In which case the meaning of the proposition is, that the force which results out of all, tends to the point S. But if any force, acts perpetually in the direction of lines perpendicular to the describ'd surface ”

“ And in like manner as in the circle, or in the ellipsis, if the forces are directed to the centre of the figure placed in the abscissa, those forces by increasing or diminishing the ordinates in any given ratio, or even by changing the angle of the inclination of the ordinates to the abscissa, are always augmented or diminished in the ratio of the distances from the centre; provided the periodic times remain equal; so also in all figures whatsoever, if the ordinates are augmented or diminished in any given ratio, or their inclination is any way changed, the periodic time remaining the same ”