Ethica II: The Lemmas on Bodies

Spinoza's physics is advanced via a series of axioms and lemmas to 2p13: "The object of the idea constituting the human mind is the body--i.e. a definite mode of extension actually existing, and nothing else."

```  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.
```

Ax1 and Ax2 together assert that to be at rest is to be doing something (rest is not the cessation of motion but a dynamical state of equilibrium); to be at rest is to be holding a place. Whether a body is moving or at rest depends on the frame of reference one happens to adopt for that body in conjunction with its surrounding physical environment. Spinoza uses this position to refute Descartes' contention that God must act so as to conserve the total amount of motion in the universe: if motion is relative, the requisite conservation laws can all be achieved through shifts in mathematical analysis.

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.