Garber's reading in 'Descartes' Metaphysical Physics' shows Descartes' reciprocity thesis directly entails that no single body has privileged mover status, which just is a claim about relational motion.
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A special position where one thing gets to be 'in charge' of motion or change, while others are passive; the claim is that no body has this special position.
Relational motion(as the alternative view Descartes may or may not be defending)
The idea that motion only exists or matters in relation to other objects or observers—something is 'moving' only because we're comparing it to something else that's stationary.
entails(describes a logical relationship between statements)
Logically forces or guarantees; if A entails B, then whenever A is true, B must also be true.
reciprocity thesis(Associated with the sufficiency claim; its denial requires an alternative account of §17)
The thesis that the conditions for objective representation and the conditions for unified self-consciousness mutually entail one another