B76 Proceedings of the Royal Society 
straiglitly and mutually proportionally in relation to any frame 
whatever. Or the two flies might be increasing their distance 
apart and afterwards diminishing it j but no approach after reces- 
sion is possible for points moving straiglitly and proportionally in 
relation to any frame whatever. 
The explanations now given are sufflcient to show that there can 
be mutual motions of various bodies, so related as to have a pro- 
perty of being uniform rectilinear mutual motions, and to explain 
the nature of that mutual relation. This is quite irrespective of 
any idea of chronometry, or any idea of absolute rest or motion in 
the universe, or of any idea of absolute clinural rest or absolute 
rotation, and of any distinction whereby one body might be said to 
be in absolute rotation and another devoid of absolute rotation. 
The mutual relation described has been purely kinematic, and will 
not be at all altered by the superposition of any new motion whether 
of translation or of rotation, the meaning of this statement being 
rendered intelligible by consideration of the attachment of the 
original reference frame to the floor and side walls of the cabin of a 
ship at sea, already mentioned. 
Now, to pass from mere geometric or kinematic motions, governed 
mutually by connecting mechanism to the motions of bodies existing 
in space free from any such governance, we are to accept as an 
established law of nature, established through multitudinous obser- 
vations and speculations, together with theories confirmed by multi- 
tudinous agreements, the following, which may be called the law of 
inertia. 
The Law of Inertia. 
For any set of bodies acted on each by any force, a reference 
FRAME and a reference dial-traveller are kinematically possible, 
such that relatively to them conjointly, the motion of the mass- 
centre of each body, undergoes change simultaneously with any 
infinitely short element of the dial-traveller progress, or with any 
clement during which the force on the body does not alter in direc- 
tion nor in magnitude, which change is proportional to the intensity 
of the force acting on that body, and to the simultaneous progress 
of the dial-traveller, and is made in the direction of the force. 
