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AXIOM I. All bodies are either in motion or at rest.

AXIOM II. Every body is moved sometimes more slowly,
sometimes more quickly.

LEMMA I. Bodies are distinguished from one another in
respect of motion and rest, quickness and slowness, and not in
respect of substance.

Proof.--The first part of this proposition is, I take it,
self--evident. That bodies are not distinguished in respect of
substance, is plain both from I. v. and I. viii. It is brought
out still more clearly from I. xv, note.

LEMMA II. All bodies agree in certain respects.

Proof.--All bodies agree in the fact, that they involve the
conception of one and the same attribute (II., Def. i.).
Further, in the fact that they may be moved less or more quickly,
and may be absolutely in motion or at rest.

LEMMA III. A body in motion or at rest must be determined to
motion or rest by another body, which other body has been
determined to motion or rest by a third body, and that third
again by a fourth, and so on to infinity.

Proof.--Bodies are individual things (II., Def. i.), which
(Lemma I.) are distinguished one from the other in respect to
motion and rest; thus (I. xxviii.) each must necessarily be
determined to motion or rest by another individual thing, namely
(II. vi.), by another body, which other body is also (Ax. i.) in
motion or at rest. And this body again can only have been set in
motion or caused to rest by being determined by a third body to
motion or rest. This third body again by a fourth, and so on to
infinity. Q.E.D.

Corollary.--Hence it follows, that a body in motion keeps in
motion, until it is determined to a state of rest by some other
body; and a body at rest remains so, until it is determined to a
state of motion by some other body. This is indeed self--evident.
For when I suppose, for instance, that a given body, A, is at
rest, and do not take into consideration other bodies in motion,
I cannot affirm anything concerning the body A, except that it is
at rest. If it afterwards comes to pass that A is in motion,
this cannot have resulted from its having been at rest, for no
other consequence could have been involved than its remaining at
rest. If, on the other hand, A be given in motion, we shall, so
long as we only consider A, be unable to affirm anything
concerning it, except that it is in motion. If A is
subsequently found to be at rest, this rest cannot be the result
of A's previous motion, for such