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1. Descartes held that all other properties arise from theconfigurations and motions of such bodies — from geometriccomplexes. See Garber 1992 for a comprehensive study.

2. Descartes’ definition is complicated by the phrase ‘andconsidered as at rest’, something perhaps added to make itconform more closely to the pre-theoretical sense of‘motion’; however, in our discussion transference is allthat matters, so we will ignore those complications

3. Additionally, as Pooley 2002 points out, just after he claims thatthe Earth is at rest ‘properly speaking’, Descartes arguesthat the Earth is stationary in the ordinary sense, because commonpractice is to determine the positions of the stars relative to theEarth. Descartes simply didn’t need motion properly speaking toavoid religious conflict, which again suggests that it has some othersignificance in his system of thought.

4. See Newton, 1999 for an up-to-date translation of thePrinciples. See Stein 1967 and Rynasiewicz 1995 and 2018for important, and differing, views on the issue; for lessons to bedrawn from both authors see Huggett 2012. For connections toNewton’s theology, see Janiak 2015.

5. Of course, there are other features of Newton’s proposal thatturned out to be empirically inadequate, and are rejected inrelativity theory: for instance, Newton’s account violatesthe relativity of simultaneity and postulates a non-dynamicalspacetime structure.

6. It is worth noting that Newton was well aware of these facts; theGalilean relativity of his theory is demonstrated in Corollary V ofthe laws of thePrincipia, while Corollary VI shows thatacceleration is unobservable if all parts of the system accelerate inparallel at the same rate, as they do in a homogeneous gravitationalfield.

7. Note that Samuel Clarke, in hisCorrespondence with Leibniz,which Newton had some role in composing, advocates the property view,and note further that when Leibniz objects because of the vacuumproblem, Clarke suggests that there might be non-material beings inthe vacuum in which space might inhere.

8. Another aspect of absolute space is its inertness: see Biener 2017for Newton’s changing views on the subject.

9. Another way of thinking about this space is as possessing — inaddition to a distance between any two simultaneous points and atemporal interval between any points — a three-place relation ofcolinearity, satisfied by three points just in case they lie on astraight line.

10. Note that fundamentally, in the metaphysics of monads that Leibnizwas developing contemporaneously with his mechanics,everything is in the mind of the monads; but the point thatLeibniz is making here is that even within the world that is logicallyconstructed from the contents of the minds of monads, space is ideal.

11. There’s a real puzzle here. Collision presupposes space, butprimitive forces constitute matterprior to any spatialconcepts — the primitive active and passive forces ground motionand extension respectively. See Garber and Rauzy, 2004.

12. However, just to muddy the waters, Leibniz also claims that as amatter of fact, no body ever has zero force, which on the readingproposed means no body is ever at rest, which would be surprisinggiven all the collisions bodies undergo.

13. Of course, the argument works by showing that, granted the differentstates of rotation, there are states of rotation that cannot merely berelative rotations of any kind; for the differences cannot be tracedto any relational differences. That is, granted the assumptions of theargument, rotation is not true relative motion of any kind.