Axiom 1: All bodies are either in motion or at rest. Axiom 2: Each single body can move at varying speeds. Lemma 1: Bodies are distinguished from one another in respect of motion and rest, quickness and slowness, and not in respect of substance. Lemma 2: All bodies agree in certain respects. Lemma 3: A body in motion or at rest must have been determined to motion or rest by another body, which likewise has been determined to motion or rest by another body, and that body by another, and so ad infinitum. Lemma 4: If from a body, or an individual thing composed of a number of bodies, certain bodies are separated, and at the same time a like number of bodies of the same nature take their place, the individual thing will retain its nature as before, without any change in its form. Lemma 5: If the parts of an individual thing become greater or smaller, but so proportionately that they all preserve the same mutual relation of motion-and-rest as before, the individual thing will likewise retain its own nature as before without any change in its form. Lemma 6: If certain bodies composing an individual thing are made to change the existing direction of their motion, but in such a way that they can continue their motion and keep the same mutual relation as before, the individual thing will likewise preserve the same mutual relation as before, the individual thing will likewise preserve its own nature without change of form. Lemma 7: Furthermore, the individual thing so composed retains its own nature, whether as a whole it is moving or at rest, and in whatever direction it moves, provided that each constituent part retains its own motion and continues to communicate this motion to the other parts.
Lem1 considers Galileo's problem: how are bodies to be distinguished from the space they inhabit; Spinoza argues (as Newton will later affirm) that bodies are not distinguished by anything other than the forces operative within them (stack this up against the "standard model" in particle physics today).
Lem2 contends that all bodies move and establish equilibria.
Lem3 is Spinoza's version of the principle of inertia: the principle is a corollary to the no-vacuum argument: bodies are an infinite totality. Spinoza regards individuals as groups of bodies nominally lumped together so as to communicate their motion to the equilibrium of the whole; the more parts a body has, the more easily it will retain equilibrium.