 As is familiar from any rotating system, there will be a tendency for the water to recede from the axis of rotation in the latter case: in (i) the surface of the water will be flat (because of the Earth’s gravitational field) while in (ii) it will be concave. The analysis of such ‘inertial effects’ due to rotation was a major topic of enquiry of ‘natural philosophers’ of the time, including Descartes and his followers, and they would certainly have agreed with Newton that the concave surface of the water in the second case demonstrated that the water was moving in a mechanically significant sense. There is thus an immediate problem for the claim that proper motion is the correct mechanical sense of motion: in (i) and (ii) proper motion is anti-correlated with the mechanically significant motion revealed by the surface of the water. That is, the water is flat in (i) when it is in motion relative to its immediate surroundings – the inner sides of the bucket – but curved in (ii) when it is at rest relative to its immediate surroundings. Thus the mechanically relevant meaning of rotation is not that of proper motion. (You may have noticed a small lacuna in Newton’s argument: in (i) the water is at rest and in (ii) in motion relative to that part of its surroundings constituted by the air above it. It’s not hard to imagine small modifications to the example to fill this gap.)